Rebounding from a soft first quarter, Garfinckel, Brooks Brothers, Miller & Rhoades Inc. yesterday reported its second-quarter profits were up 11.5 percent over last year and sales increased by 10.1 percent.

Garfinckel's sales for the 13 weeks through Aug. 2 climbed to $97.5 million from $88.6 million last year, and earnings increased to $930,000 (20 cents a share) from $834,000 (18 cents) a year ago.

The diversified retailer's speciality divisions, particularly Brooks Brothers, produced most of the gain in sales, and higher margins and tighter inventory controls aided profits, said Chairman David R. Waters.

The Garfinckel corporation was the second big Washington retailer to report a second-quarter upswing in business. On Monday, Woodward & Lothrop reported its quarterly earnings were up 10 percent from the year before.

But like Woodies, Garfinckel's said its second-quarter gains did not offset completely the dismal performance in the first three months of the year, so half-year sales and profits were off. And like Woodward & Lothrop Chairman Edwin Hoffman, Garfinckel's Waters said the second quarter's improvements did not assure that business would be good the rest of the year.

Waters said he "views the remainder of the year with caution," pointing out that the company "faces the same economic problems confronting the industry generally, including the on-going recession, an extremely competitive retail environment and sharply higher interest rates."

For the first half of the year, Garfinckel's reported earnings from continuing operations of $2.04 million (45 cents) down from $2.47 million (54 cents) in the first half of fiscal 1979. Last year the company also showed a $3.2 million nonrecurring gain from the disposition of its Jospeh R. Harris division, which added 70 cents a share to earnings for the six month period.

Washington Gas Light Co. reported its common stock earnings in increased by a dime to the equivalent of $2.63 a share for the 12 months ended July 31.

Total Washington Gas profits climbed to $14.3 million from $13.9 million; after paying preferred stock dividends, the company's profits amounted to $11.5 million, up from $11.03 million in the same period a year ago.

For the 12-month period, natural gas sales declined by about 1 percent, but gas company revenues climbed to $418.5 million from $355.4 million.

In other earnings reports yesterday, Virginia First Savings and Loan Assn., headquartered in Petersburg, posted a profit of $1.2 million ($1.12 a share) for the fiscal year ended June 30, down from $1.6 million ($1.53) last year.

Assets of the association increased to $216 million at year end and will climb to more than $280 million with acquisition of United Savings and Loan Assn. of Vienna, the company said. Applications to take over United and its six branches in the Washington suburbs are now awaiting approval.

Another Virginia business, Stuart McGuire Co. of Salem plunged into the red for the first six months of the year, posting a loss of $798,000 in contrast to earnings of $390,000 (40 cents a share) last year. Revenues for the first half were up 31 percent to $28,7 million from $21.9 million.