The South Korea carmakers conceded on Friday that they had overstated the fuel efficiency ratings on more than 1 million recently sold vehicles in the United States and Canada, and agreed to compensate owners for the additional fuel costs.

The lawsuit, which seeks class-action status, was filed in the U.S. District Court for Central California on behalf of 23 Hyundai and Kia car owners and challenges the automakers' compensation plan.

Under the plan, customers will receive a debit card that will reimburse them for the difference in fuel economy, and an extra 15 percent to the amount will be added to acknowledge the inconvenience.

The lawsuit, which like a separate one filed in the Southern District of Ohio this week, is seeking further compensation for the reduced value of their vehicles.