Bank of Virginia Co., which owns the state's third-largest bank, reported income before securities gains or losses of $6 million ($1.05 a share) in the second quarter, up from $4.1 million (73 cents) in the second quarter of 1981.

The bank said it lost $848,000 buying and selling securities, leaving net income of $5.2 million (90 cents) compared with $3.2 million (57 cents) in the three months ended June 30, 1981.

For the first half of the year, Bank of Virginia earned $10.5 million ($1.85) before securities transactions, up from $6.8 million ($1.23) in the first half of last year. After securities gains and losses, the bank holding company earned $8.2 million ($1.43) compared with $6 million ($1.07) during the first half of last year.

Avemco Corp., a Bethesda insuror of aircraft, reported second-quarter net income of $630,000 (25 cents a share) compared with $800,000 (32 cents) last year as revenues rose to $8.2 million from $7.9 million.

First-half net income was $2 million (79 cents) compared with $1.9 million (78 cents). Revenues were $16.8 million compared with $14.9 million.

Two factors that affected the second-quarter results were a decrease in short-term investment yields and increased insurance losses caused by a larger number of liability claims, the company said.

STSC Inc., a Rockville corporate financial management systems and services firm, reported a loss for its fiscal year ended May 31 of $1.8 million compared with earnings of $1.8 million (94 cents a share) last year. Revenues were $27.8 million compared with $27.6 million.

For the fourth quarter, the company reported a loss of $1.4 million compared with net income of $491,000 (22 cents) last year. Revenues were $6.5 million compared with $7.1 million. Results included a $1.7 million write-down of certain software and hardware assets.

The company said fourth-quarter earnings were affected by the cost of moving its headquarters to Rockville from Bethesda last October and the expansion of computer capacity.

Syscon Corp. officials reported that its second-quarter earnings increased to $631,000 from $562,000 last year. Per-share earnings dropped from 25 cents to 24 cents because more shares were outstanding.

Revenues for the D.C.-based systems engineering and software products company increased during the period to $17.3 million from $14.9 million last year.

For the half, Syscon profits were $1.07 million (40 cents a share) compared with $1.06 million (46 cents). Revenues increased to $33.7 million from $29.4 million. Syscon officials said that earnings for the half were reduced by 7 cents per share because of an arbitration ruling the company is contesting.

First & Merchants Corp., a Richmond holding company for Virginia's fourth largest bank, more than doubled its second-quarter earnings to $8.4 million ($1.27 a share) from $3.9 million (61 cents) last year. After securities transactions, profits were $8.1 million ($1.23) compared with $3.6 million (57 cents).

In the half, F&M profits grew to $14.03 million ($2.11) from $8.2 million ($1.26) last year.