Government Employees Insurance Co., a large auto insurance firm that was close to insolvency in mid-1976, yesterday reported its second consecutive profitable quarter.

Unaudited figures released by chairman John J. Byrne showed overall profits of $8 million in the fourth quarter of 1976 compared with a net loss of $107.2 million for the same three months in 1975.

Despite a continued loss from underwriting auto insurance policies in the quarter of $4 million, compared with a loss, of $117.3 million in the 1975 period, Geico was able to remain profitable because of $9.9 million of income from investments and capital gains of $2.8 million in the recent quarter.

Geico ended the year with a policy-holders' surplus of $136 million, a sharp recovery from $49.3 million at the end of 1975, reflecting a successful offering of new preferred stock last November which replenished depleted reserve funds.

Because of losses earlier in 1976, Geico suffered a net loss for the full year of $26.3 million compared with a record loss $126.5 million in 1975. The firm's loss on underwriting auto policies dropped to $60 million from $191 million.

In an interview last night, Byrne noted that the fourth-quarter loss from underwriting auto insurance actually was $7 million before taking into accounts an industry reinsurance agreement with other companies, reached last year, under which the loss is spread out among the insurance companies involved.

"The loss is completely unacceptable althought there is a hell of a nice trend toward reduced underwriting losses . . . by no means are we operating the company in a satisfactory manner but the $8 million profit helps us draw another deep breath," Byrne stated.

He expressed concern that Geico's premiums written in the recent quarter totaled $118 million, the smallest quarterly volume in many years and down from $161.1 million for the same periods of 1975. The decline in business reflects both a deliberate Geico effort to eliminate unprofitable customers and voluntary withdrawals by other customers - for an overall decrease of some 800,000 policyholders in the past year, Byrne said.

To keep preferred customers - those with the safest driving records - Geico recently won approval to reduce some rates in Maryland and is seeking similar reductions in Virginia and the District. Byrne said he does not expect Geico to request any significant rate increases in the near future.

Most of the business decline in the recent quarter occurred in October and November, Byrne said. "We are going to try to retain our good customers," he added.

Audited results will be included in Geico's annual report, scheduled to be mailed in early March. Byrne is scheduled to address securities analysts in New York on Friday, to describe Geico's recovery program.