The FCC had asked Verizon Wireless last year about $1.99-a-megabyte data access fees that appeared on the bills of customers who didn’t have data plans but who accidentally initiated data or Web access by pressing a button on their phones.

In a statement on its website Sunday, Verizon Wireless said most of the 15 million customers affected will receive credits of $2 to $6 on their October or November bills. Some will receive larger sums. Customers no longer with the New York-based carrier will get refund checks.

“Verizon Wireless values our customer relationships and we always want to do the right thing for our customers,” said Mary Coyne, deputy general counsel for Verizon Wireless.”The majority of the data sessions involved minor data exchanges caused by software built into their phones; others involved accessing the Web, which should not have incurred charges. We have addressed these issues to avoid unintended data charges in the future.”

Verizon has said that it stopped charging such fees when a customer started using a data service but then quickly shut it off.

Verizon Wireless, the largest cell phone carrier in the U.S., is a joint venture between Verizon Communications Inc. and Britain’s Vodafone Group PLC.