WASHINGTON -- The women’s group that gave Mitt Romney binders of women’s resumes contested his version of the story on Wednesday, while Democrats worked to criticize his record, although Romney apparently named a comparable percentage of women to top spots as President Barack Obama.

In response to a question about unequal pay for women at Tuesday’s debate, Romney said that as governor of Massachusetts, he made “a concerted effort” to find qualified women to serve in his Cabinet.

“I went to a number of women’s groups and said, ‘Can you help us find folks,’ and they brought us whole binders full of women,” he said.

That’s not exactly what happened, according to the nonpartisan Massachusetts Government Appointments Project, or MassGAP. Romney didn’t request the binders, the group said in a statement Wednesday.

In the lead-up to the 2002 gubernatorial election, MassGAP approached the campaigns of both Romney and his opponent, Shannon O’Brien, asking them to “make best efforts” to ensure that the number of women appointed to state leadership positions was proportionate to the population of women in Massachusetts, which was about 52 percent at the time. Both candidates pledged to do so and agreed to work with MassGAP during the process.

After the election, MassGAP presented Romney’s transition team with binders containing hundreds of resumes for top applicants vetted by the group.

By the middle of Romney’s term in 2004, women made up 42 percent of new appointments to senior-level positions in Massachusetts government, according to MassGAP. But by the end of his term in 2006, the percentage of newly appointed women in those positions had dropped to 25 percent, the group said.

“Romney paid lip service to the public about hiring more women in senior positions and treated it like a quota,” Jesse Mermell, a former executive director of MassGAP, said in a press call organized by the Obama campaign. “But like with so many other things with Mitt Romney, the facts didn’t match the rhetoric. Facts are facts, and despite what Gov. Romney claimed in the debate last night, there were fewer women in senior administration positions during his term than the governors who came before and after him.”

The Romney campaign countered with a message from Kerry Healey, who served as lieutenant governor with Romney when he was governor of Massachusetts.

“In fact, of the twenty top positions in the Romney administration, ten of them were filled by women, more than any other state in the nation,” Healey wrote. “Romney’s Chief of Staff was a woman – Beth Myers. As we took office, our administration actively sought to recruit the best and brightest women the Commonwealth had to offer. And Governor Romney wasn’t just checking a box. He sought out our counsel, and he listened to our advice. We didn’t always agree, but we were always respected. Mitt Romney didn’t judge the people who were in his administration by their gender. He wanted the best, male or female.”

A Romney spokeswoman said the governor worked with MassGAP to find the best-qualified women for top government posts. “The efforts resulted in Massachusetts having the most women in top positions in the entire country,” Andrea Saul said.