These are bittersweet days at the Government Employees Insurance Co. John J. Byrne, the corporate folk hero, is leaving the company he rescued, taking with him his quick wit, his talent for office pranks and his ability to amuse and amaze.

Before announcing his departure, Byrne put Geico's future in the hands of two men he brought in several years ago to help run the company: William B. Snyder, 56, head of the insurance operation, and Louis A. Simpson, 48, who oversees Geico's investments. Until now, the three have been a team -- Byrne, the dynamic and colorful cheerleader; Snyder, the cost-conscious insurance operator, and Simpson, the high-performance investment expert.

Now Snyder, who came to Geico eight years ago, has moved into Byrne's seat as chairman and chief executive officer of the parent company, Geico Corp. Simpson, who has been on the job for six years and is Geico's chief financial officer, has become vice chairman.

Byrne and Snyder worked together for 17 years, starting at the Travelers Insurance Cos. in Hartford. "I'll miss not having him around," said Snyder. "He is more inventive, more audacious. He comes up with all the good ideas, and I figure out what's wrong with them."

Not much will change immediately at Geico -- except perhaps the level of fun in the executive suite, according to the collective view of Snyder, Simpson, Byrne, investor Warren Buffet of Omaha, who owns a 36.6 percent stake of Geico, and several Geico watchers. "The basic business is not going to change in any way, shape or form," Snyder said.

Byrne's new job will make him chairman and chief executive officer of Fireman's Fund Corp., which will be spun off by its parent, American Express Co., through a public offering of stock. Fireman's Fund Corp. handles $4 billion in premiums, of which 70 percent are commercial. American Express will continue to sell health and life insurance through its subsidiary, Fireman's Fund American.

Byrne accepted the offer from American Express while Snyder and his wife were vacationing in Europe. Byrne called Snyder in West Germany last Tuesday to break the news. The Snyders cut short their vacation, hopped a plane home and on Wednesday, at the Geico board of directors meeting, Snyder was elected as the new boss.

The way Snyder remembers that call from Byrne, it began with Byrne saying, "You're going to be surprised." Byrne reminded Snyder that he had been talking with American Express about a possible insurance deal between Fireman's Fund and Geico. Then Byrne said, "They've made me an offer to run the company that I can't refuse." Byrne also said he would recommend that the board of directors name Snyder to take his place.

Snyder said he told Byrne that it was "a very dumb thing" to be going out to California (headquarters of Fireman's Fund) when he already had such a good job. Byrne responded that Snyder and Simpson were doing such a good job running Geico that, "You don't leave me anything to do." And Byrne added, "If I hang around, I'll just screw it up."

Snyder said that he'd heard Byrne's "I-haven't-got-anything-to-do-line" before. Byrne frequently told investment analysts that, with Snyder running insurance and Simpson handling investments, he no longer had much to do. That wasn't quite true, Snyder noted. Although Byrne wasn't involved in the day-to-day operations, he had a full schedule, reviewing possible acquisitions and representing the company on industrywide problems.

One close observer noted, however, that Byrne had seemed to be "getting restless." Another said he thought Byrne had not been very successful on the acquisition front.

For Snyder, his ascent to the top of Geico is an event he could not have anticipated when he was young. Snyder grew up in Clarksburg, W. Va., where he remembers the freshly laundered sheets on the clotheslines turning gray from the soot in the smoke of the soft coal that was mined and burned nearby. His father died when Snyder was a child, leaving a son and seven daughters. When his mother was well, Snyder recalls, she did housework, or worked as a seamstress to support her children. When she was ill, he said, the family survived on welfare. "Those were very poor, tough days for our family," he said.

With little hope of going to college -- no one in his family had ever gone -- Snyder enrolled in a commercial course in Washington Irving High School. But an perceptive teacher, Maud Yoak, recognized a special talent in Snyder and steered him into a college curriculum. She also helped him get a $200 scholarship to pay his tuition at Davis and Elkins College in Elkins, W. Va. Working at odd jobs to pay his way through school, he joined the Naval Reserve Officer Training Corps, which sent him to school at the University of South Carolina. Part of the program was an eight-week summer cruise. "I got seasick on every ship the Navy had except a submarine. And I decided the Navy was not going to be my life."

Yoak's role in putting him on the path to college and his acceptance by the Air Force for its pilot training program were turning points in Snyder's life. He later finished college at Texas Technological University in Lubbock, getting a degree in business.

The Air Force taught him self-confidence as he survived first the rigorous training program, then 12 combat missions in B-29s over Korea. He came out of the service believing he could do anything he set his mind to.

"Success is the progressive realization of an ideal," Snyder is fond of saying. He defines it as a method for achievement: First, pick a goal, work toward it and achieve it. Then pick a new and higher goal, work toward it and achieve it.

"That's been the story of my life," said Snyder, who climbed the corporate ladder rung by rung over 30 years. "I had no grand aspirations. I didn't plan it. . . . I worked hard day-to-day at whatever job I had."

Coming from a youth shadowed by poverty, did Snyder ever expect to rise so high? "No, frankly I never thought about it." Indeed, he had been happy in his job at Travelers Insurance and was happy with each successive role he was given at Geico. Even now, sitting at the top of the heap, Snyder sounds like he's telling the truth when he says he would have been "just as happy" if Byrne had stayed at Geico for another 10 years.

Success has brought its rewards. Snyder says he earns about $300,000 a year and is eligible for $50,000 to $70,000 a year in cash bonuses. Simpson's salary is similar.

Snyder routinely works a six-day week, spending Saturdays in the office. His office and Byrne's, which occupy a 75-foot-long suite on the eighth floor of the Geico building in Chevy Chase, are connected by a conference room.

There is no executive dining room at Geico. It was one of the first luxuries to go when Byrne took office in 1976. Visiting dignitaries, along with company officials, go through the cafeteria line like everyone else.

That style seems to fit Snyder's method of doing business. Buffet says Snyder is the kind of executive who "cares about getting that last one-tenth of 1 percent."

Watching the other end of the business is Simpson. When D.C. officials held a hearing recently to discuss Geico's request to increase rates for D.C. customers by an average of 8.5 percent, one official held up a copy of a recent newspaper column, which cited Simpson's 21.26 percent annualized return on Geico investments for 1982, 1983 and 1984. The official suggested that the article proved Geico was doing well enough and didn't need the additional money. Indeed, Simpson's record in managing $1.5 billion of Geico's money has drawn wide attention in the Washington investment community.

Simpson, a native of Chicago, graduated from Ohio Wesleyan University and earned a master's degree in economics from Princeton University. Before coming to Geico, he was president and chief executive officer of Western Asset Management, a subsidiary of Western Bancorp. in Los Angeles. An urbane money manager who favors cigars, Simpson came to Geico six years ago when the company's portfolio was about $900 million, with most of the money invested in tax-exempt and corporate bonds. He moved much of Geico's investments into common stocks, where he believed there were opportunities for maximizing returns.

Even in the midst of his investment success, Simpson is apt to worry about what happens next. Writing in Geico's 1984 annual report, Simpson reported a 15.18 percent portfolio increase compared with a 6.25 percent rise for the Standard and Poor's 500. But he cautioned stockholders, "I am quite apprehensive about our ability to duplicate these outstanding results in the future."

Geico's new team will have its share of challenges.

Geico, with total revenue of $995 million last year, including $800 million in premiums, is one of the nation's more successful property-casualty insurance companies, ranking about ninth, with about 2 percent of the market.

Earnings have been flattening out, however, and Snyder expects a flat year in 1985. The root of the problem goes back to 1983, when the company decided to hold down rates to attract new business. New business came flowing in, but unexpectedly, in mid-1984, the frequency and severity of accidents rose sharply, cutting into Geico's underwriting profits. The episode found the company running about six months behind on car rate increases. The company now is trying to catch up on rate hikes and cope with growing customer claims produced by the greater amount of business.

Byrne called the leveling off of profits a "temporary blip" and, with his tongue firmly in his cheek, said he engineered it that way "so my associates could say what a great turnaround they had. . . ."