One day last winter when Kathryn Herman was living in Virginia Beach, she returned home from work to find her two sons sitting in the snow.

The babysitter, fatigued by their company, had locked them outside.

That evening, she recalled, she decided to pack her bags and return to Arlington.

"I was tired of shuttling my children from one irresponsible babysitter to the next. I was tired of hearing, 'Gee, lady, I'm really sorry, but you're going to have to find a new sitter,' two days before she left.

"I knew that if I moved back to Arlington, I could place my children in the extended daycare program."

Herman, a single mother, joined nearly 100 working mothers and Northern Virginia officials last week at a public hearing on extended day care at Northern Virginia Community College in Annandale.

Many attending the hearing said they hoped the General Assembly would pass legislation allowing extended day-care service in public schools.

Last year, Falls Church and Arlington school officials were notified that their programs might be eliminated next June.

These programs and one in Charlottesville are the only three extended day-care programs operated by Virginia public schools.

Unless a bill passed by the General Assembly last session is amended, the programs will have to be transferred to another agency or eliminated by June 1981.

The bill arose from a court ruling that school systems did not have written authority to conduct day-care programs.

About 1,200 elementary school children are enrolled in the 10-year-old Arlington program. They attend only during the school year, from 7 a.m. to 6 p.m. Fees are based on a sliding scale, from $7.90 to $77 a month, depending on a family's ability to pay.

The Falls Church program, which serves 105 students, operates year-round.

"I was so relieved when my kids enrolled in the day-care program," Herman said. "I didn't have to worry about my son lighting my living room on fire when I wasn't there. I didn't have to worry about them being locked outside.

"But most importantly, my kids love it. If I'm not ready to go at 7:30, they literally shove me out the door.

"I don't know what I'm going to do if the program is eliminated," said Herman in exasperation. "I have to work. There is no one who is going to support me and my two children."

The Arlington and Falls Church programs are surviving through the grace of a 16-month extension.

At the time the extension was granted, the state division for children was directed to form a task force to study school-day-care-programs. The task force was told to report to the General Assembly in November.

The task force liked what it saw. Members of the panel say they plan to recommend that the General Assembly approve legislation giving schools authority to operate such programs.

Parents and local governments would pay for the programs. In Arlington, school officials reported that parents pay 70 percent of the costs.

"I don't think anybody questions the need for day care," said Ron Thompson of the state division of children, who helped direct the task-force study. "The question is whether the public or private sector should provide it. The committee feels schools are one way of meeting the need."

Parents and school officials interviewed heartily agreed with the panel's findings, saying that the school-day-care programs were the ideal answer for school-aged children with working parents.

Most said private programs were unwilling to take children on a part-time basis and that transporting children from school to another center was a major problem.

"In Northern Virginia, dependable extended day care for school-aged children is negligible," said Falls Church resident Beverly Rausch. "The majority of women in my neighborhood work, and the only teenager near my home has many after-school activities of her own and is not, therefore, available for after-school sitting."

"Without our school system's extended day-care program," Rausch continued, "employment for me would be impossible."

Fairfax County and Alexandria also have extended day care programs in the schools, but outside organizations operate these programs.