“Since it occurred over a long period of time in relatively small amounts, the district was unaware of these discrepancies until it instituted new financial controls specifically related to the collection and depositing of cash in the cafeterias,” Michael Horyczun, spokesman for New Canaan Public Schools, said in an email Monday night.

The New Canaan Board of Education, which governs public education in the town, filed a complaint with the police in December 2017. The women resigned under threat of termination the same month, according to their arrest warrants.

Ms. Wilson worked as the assistant food director at New Canaan High School, which reported thefts at the cafeteria totaling about $350,000 from 2012 to 2017, the police said. Under her supervision, cashiers did not count the money at the end of the day, but rather Ms. Wilson did that task herself, the police said.

Her sister, Ms. Pascarelli, was in charge of the food program at Saxe Middle School, which lost about $127,000 in the same five-year period, the police said. Co-workers told the police that Ms. Pascarelli came to the cash registers and removed large bills in between lunch periods.

“Look, you’re caught,” the police told Ms. Pascarelli during an interview, according to her arrest warrant.

Ms. Pascarelli told the police that her former co-workers were lying. “I would never take money,” she said. “I know better than that.”

But after Ms. Pascarelli left her job at the cafeteria, the police said, the average daily cash intake in the middle-school cafeteria — which once averaged below $40 — jumped up to about $150.