Volkswagen admitted to installing software designed to cheat emissions tests in about 11 million cars worldwide. While the cars passed emissions tests, they released nitrogen oxide up to 40 times the level allowed by federal law. Here's what you need to know about the scandal. (Jorge Ribas/The Washington Post)

The deception perpetrated by Volkswagen in the United States reaches around the globe, with about 11 million cars worldwide equipped with software designed to cheat emissions tests, the company said Tuesday.

The automaker said it will set aside $7.3 billion to cover fixes and “other efforts to win back the trust of our customers.” That amount is likely to fall many times short of the actual costs, including car repairs, lawsuits and government penalties around the world.

Exactly what alterations are necessary on all of those cars is unknown, and independent engineers said it could be extremely difficult to repair the emissions systems without harming engine efficiency and performance. None could offer what they deemed a reliable estimate of the cost of a potential repair.

“In my German words, we have totally screwed up,” Volkswagen’s U.S. chief, Michael Horn, said at an event in Brooklyn late Monday night.

The broad scale of the deception suggests that knowledge of the emissions cheating was widespread, and Justice Department investigators are focusing on the actions of executives, according to two people familiar with the inquiry. German news outlets reported Tuesday that the firing of chief executive Martin Winterkorn is imminent, citing unidentified members of the company’s board.

It’s not clear what alterations are needed in the Volkswagens that have the cheating software. (Morris Mac Matzen/Reuters)

Also Tuesday, new details of the cat-and-mouse interactions between suspicious regulators and the German car giant showed how far the company was willing to go to assure the government that, contrary to the best evidence, nothing was amiss in its diesel cars. Last year, Volkswagen informed regulators that it was initiating a 500,000-car recall in the United States that would fix the problem. The recall was either a technical failure or, as some U.S. officials said, a ruse.

Whether those involved in the emissions cheating software will face more severe penalties is unknown, but anger among customers, who are stuck with cars that violate pollution standards, and dealers, who are left with unsold inventory, has become increasingly evident. Their appeals have been heard in Washington.

“It is an outrage that VW would take advantage of its consumers by purposely deceiving them on their mileage on diesel vehicles. . . . There ought to be some prosecutions, and corporate executives that knew this and have done it ought to be going to jail,” Sen. Bill Nelson (D-Fla.) said in a speech on the Senate floor Tuesday, citing the repeated failures of automakers. “And I lay this not only on the corporate culture, I lay it at the feet of the U.S. regulatory agencies who ought to be doing their job, ought to be doing it in a forceful way.”

Noting that Volkswagen had been accused of similar tactics in the United States in the early 1970s, when the company paid fines of $120,000, Clarence Ditlow, director of the Center for Auto Safety, argued that financial penalties are not enough to keep the company honest.

“The only way to change auto company behavior is to put the responsible executives in jail,” he said.

The company’s diesel troubles have emerged over just a few days, but they were more than a decade in gestation and have roots in the company’s efforts to conform to stricter U.S. emissions standards.

Those regulations were finalized more than 15 years ago, and after a years-long grace period, were phased in between 2004 and 2007. At the time, the government acknowledged that meeting the emission standards for NOx, or nitrogen oxide, and soot would be difficult.

“Manufacturers have expressed concerns that dies­el-fueled vehicles would have difficulty meeting NOx and particulate matter levels like those contained in today’s rule,” Environmental Protection Agency officials wrote in issuing the new standards. “Clearly, these standards will be challenging.”

For a while, Volkswagen and other automakers stepped back from the U.S. diesel car market. It was a time to retool. Experts said the challenge of making a diesel engine clean enough for the U.S. standards — without compromising how the engine works — is very tough.

In 2008, Volkswagen came back with a diesel version of the Jetta. The company touted it as an example of “clean diesel.” It won raves, including “Green Car of the Year” at the Los Angeles Auto Show.

“After a three-year break that seemed interminable to fans, Volkswagen’s back in the U.S. diesel-car market with a clean-burning diesel in its popular Jetta compact sedan,” USA Today told readers.

But what seemed like a breakthrough seems now, in light of the recent revelations, like a massive cheat.

The engineering hurdles in controlling diesel emissions are high enough, some experts said, that they may have provided an incentive for companies to skirt the rules.

“I can’t say I’m surprised,” John M. DeCicco, a research professor at the University of Michigan Energy Institute, said of the revelations. “The temptation to game the system with a defeat device is definitely high because of the technical challenges.”

The company’s two-liter diesel engines in the United States were equipped with a “defeat device” that allowed the cars to pass federal emissions tests, despite the fact that while on the road, they emitted more than 10 times the permitted amounts of NOx. Those emissions help lead to smog on hot summer days, triggering a variety of health problems, particularly for asthmatics and other people with breathing difficulties.

As has been widely reported, the emission problems were detected in a May 2014 report by the International Council on Clean Transportation and West Virginia University.

A closer examination of the public documents around the case, however, shows that as recently as December, the company sought to convince government regulators that the excess emissions could be addressed with a simple software change.

At a meeting on Dec. 2, Volkswagen proposed a “voluntary recall” of about 500,000 vehicles. Officials from the United States and California, which generally takes a lead role in emissions issues, approved the plan.

“Based on this meeting, VW initiated a voluntary recall. . . . This recall was claimed to have fixed among other things, the increased real world driving NOx issue,” according to a recent letter from the California Air Resources Board to the company.

The actual extent of the recall, which is discussed in letters between government and company officials, is unknown. Company officials wouldn’t say this week how many recall notification letters Volkswagen issued to consumers or how many consumers brought their cars in for the purported fix.

But by May of this year, California officials had run road tests on cars altered under the recall. Despite the fix, the cars showed higher-than-expected NOx emissions. California engineers also created a special dynamometer test, apparently to address company concerns that the previous testing was unfair.

The California Air Resources Board “has determined that the previous recall did not address the high on-road NOx emissions, and also resulted in the vehicle failing certification standards,” it told the company this month.

“They basically ran out of excuses,” said Stanley Young, spokesman for the Air Resources Board. “They would say the tests weren’t at the right temperature or some other issue. We had them in [to our offices] several times.”

EPA Administrator Gina McCarthy said Tuesday that the agency will step up efforts to look for cheating by other manufacturers, though she suggested that Volkswagen was an “outlier” in the extent of its effort to defeat emissions standards.

“We are not going to sit around and worry about whether or not there are other devices. We’re going to find it,” she said. “So right now we’re upping our game in terms of going out and doing the work to take a look at what we need to do with other vehicles.”

McCarthy, speaking to reporters in Washington, acknowledged that the Volkswagen software ruse was “particularly difficult for us to detect. We haven’t found similar types before, but we’ll take a look and make sure we’re attacking it successfully.”

Such efforts may protect consumers in the future. For now, many of those diesel Volkswagens present a problem for their current owners, as well as for the 650 U.S. dealers with whom thousands of the suspect vehicles now sit.

Jack Fitzgerald, chief executive of Fitzgerald Auto Malls, a dealership chain that sells Volkswagens in Annapolis and Frederick, Md., said he is awaiting details from the company about exactly what is needed to fix the problem.

“They haven’t given us any idea what the fix is,” Fitzgerald said. “All we know is that we have parked the cars that have not sold, and we can’t deliver them, and we’re waiting for further information from Volkswagen.”

Tom Doyle, manager at Wheaton Service Center, an independent shop that specializes in Volkswagens, said that customers “just feel slightly betrayed by Volkswagen.”

He’s talked about little else with them since the news broke Friday of the EPA’s allegations.

“They’ve called me wanting to see what kind of drop in fuel economy they’re likely to see once a fix is applied,” he said. “The thing that frightens me is that they thought they could get away with it. . . . Every one of those half-million cars that they sold has the evidence onboard.”

Moore is a freelance writer. Joby Warrick contributed to this report.