Government Employees Insurance Co., which in mid-1976 was on the brink of insolvency, continued to show profit gains for the third straight quarter.

According to figures released yesterday, net income during the first quarter of 1977 was $9.4 million compared with a loss of $26.4 million during the same period last year.

First quarter income for the locally based auto insurer was also above the $7.98 million for the last quarter of 1976.

Expressing satisfaction with the results, Geico Chairman John J. Byrne said in an interview last night that the company is "generally on track."

Byrne said, however, that he was troubled that Geico continued to lose money on auto insurance underwriting, which is figured by the conservative accounting methods required by insurance regulators. But he predicted that the $2.3 million underwriting loss would be wiped out by the end of the year.

In 1976, automobile underwriting losses fell progressively in each quarter to $3.7 million in the final three months, creating a total loss for the year of $60 million against a loss in 1975 of $191 million.

Byrne noted that the surplus insurance companies must maintain to pay off policyholder claims was $142.3 million, the highest ever for the company. A year ago during the same period, the surplus was only $39.3 million.

As part of its recovery, Geico had to boost premium rates. Now, said Byrne, "our competitors have raised prices so that our prices are getting more attractive."

He said that more than 80 per cent of the Geico policyholders who have been offered renewals have accepted. That is a "considerable improvement" over the recent policyholder attrition rate, Byrne said.