Senior officers of Government Employees Insurance Co. are hard at work on plans for establishing a holding company that would open up new lines of business for the Washington-based auto insurance firm.

At the same time, Geico is plotting an aggressive nationwide marketing program for its existing auto and homeowners insurance policies. A major new element in this plan will be "sponsorship" of Geico by trade and professional associations in direct mail advertising to the members of selected groups.

These twin programs are being developed as Geico, rebounding strongly every quarter after a debacle in 1976 that almost resulted in insolvency, finds its capital base growing faster than new insurance business.

As a result, Geico faces a prospect of having substantial amounts of money to invest in the next year and thereafter. Moreover, Geico officers want to diversify overall operations to such an extent that company operations won't be so adversely affected in the future, from a downturn in profits from auto insurance-until now vitually their single business line.

Geico Chairman John Byrne and Senior Vice President William Snyder outlined their strategy in an interview, as the company published detailed financial results for the third quarter and first nine months of 1978.

Although Geico's overall earnings increases had been reported earlier, the report this week showed that nine-month operating profits soared to $441 million from $26.9 million in the same period last year.

After accounting for 1978 investment lossis of $7.3 million (vs. investment gains of $1.1 million in the 1977 period), earnings before tax benefits related to previous losses totaled $36.8 million compared with $28.1 million.

"During the past nine quarters,Geicohas made a complete financial recovery and along the way reported rapid escalation of profits," Byrne noted in a cautionary warning about future earnings growth.

"With the stabilization we have worked so hard to achieve now accomplished, you can expect to see more moderate growth in profits from quarter to quarter as our financial results return to a more natural status."

In addition, Bryne conceded that new business sales generally are below expectations of a year ago even though they are up substantially. Geico's written premiums volume was down 5 percent for the first nine months and 2 percent in the third quarter.

Geico cut off all sales efforts at the end of 1975 when financial difficulties surfaced and the company had to trim its business base. Said Byrne: "I didn't realize it would take as long as it has to fill the pipeline again."

Inquiries about Geico policies and rates have soared to some 240,000 per quarter compared with about 100,000 a year earlier - about on target. But the company has not been able to translate as many of these inquires in actual new policies as planned.

Geico officers had wanted to sell 250,000 new policies on an annual basis and the rate of new sales in the third quarter was 43,000. At the same time, an unusually high percentage of current customers are renewing their Geico policies (some 91-92 percent) "a very pleasant surprise" since the firm had forecast a renewal rate of about 88 percent, still above industry averages.

Although Snyder a marketing specialist is embarking on an ambitious program of boosting sales of new Geico policies, Bryne said that "in all probability, we will have much more capital than Snyder is able to effectively put back in the marketplace . . . no matter how fast we grow in market, growth in capital [indicates] we must get tthe capital to work."

Therefore, Geico expects to have for its stockholders before the end of 1978 a proposal to form a one-company holding company - a new firm that would own all shares of Geico in a transaction that would require regulatory agency approval.

Geico's proposed general purpose parent company could acquire other firms, make investments or start new businesses. While Geico would be provided all the capital it needs to protect policyholders and underwrite expansion. Byrne said there may be an initial $50 million available to add other businesses.

For the time being, Byrne said he expects that Geico's substantial investments in three sister firms (Criterion, Government Employees Life and Government Employees Financial) would remain just that, rather than making those firms subsidiaries of the holding company.

In October, Geico offered to buy for cash additional shares of the life insurance firm and of Criterion, an automobile insurer which accepts customers considered high risks than those acceptable by Geico itself.

Top candidates for new business of the Geico holding company would be "anything in the financial services area we feel that can be marketed successfully through direct mail," Byrne stated.

Examples could be market-related mutual funds without commission rates, leasing, specialized investment services and supplemental retirement investment plans. The "number one value objective" is to increase stockholders' return on capital, the Geico chairman added.

Meanwhile, Snyder is concentrating on the firm's other top goal - expanding Geico's business.

An advertising campaign that focuses on Benny Goodman has helped bring Geico's name back before the public and enhanced the spirit of company employees.

But Geico's traditional method of sales is still the foundation - direct mail advertising, which eliminates sales commissions. A simple postcard, on which potential customers ask price quotations from Geico, is still the single most effective marketing tool in the company's arsenal.

The auto insurer wants to expand its business in the markets it regards as best - D.C., California, Virginia, Texas, Georgia and northern Florida. Geico will continue to focus on government workers and will add pitches to membership of various professional groups - all based on insurance value for lower premiums than most competitors.

Byrne said the major factor in keeping premiums low is worker productivity. The firm has one of the best records of productivity in the industry, measured by the amount of money spent to acquire and renew policies. But Byrne said other companies are moving in the same direction and Geico's edge will be maintained only by squeezing costs to the lowest possible point.

"We've got to return with a vengeance to the original mission of Geico - good value," Byrne stated.

Geico also has a new products task force studying such possible lines as recreation vehicle and motorcycle policies. And Snyder emphasized that Geico will continue to boost its homeowners policies sales, which were up 246 percent in the third quarter from last year but still account for only about 5 percent of overall volume.