During the second presidential debate Tuesday at Hofstra University, Mitt Romney made a remark that has emerged as the most notable gaffe of the evening when he said he had culled "binders full of women" to make sure there were women in his cabinet as governor of Massachusetts.

Social media outlets erupted at the time of the comment, — see some Twitter reactions below — but a day later, it seems Romney was a bit too quick to take ownership of the initiative in the first place.  

The Washington Post blog The Plum Line included input from Liz Levin, the chair of a women's group Romney said he had pushed his administration to reach out to in finding female applicants.

Levin told the Post's Greg Sargent that the exchange didn't happen quite Romney framed it Tuesday.

In fact, Levin tells me, the groups initiated contact with him and urged him to hire more women — when he was still a candidate — and began creating the binders themselves on their own initiative before he took office. In fairness to Romney, she says, he did agree to work with them.

...At that time, the groups, on their own, were compiling lists of qualified female candidates for positions in state government, she says. After he was elected governor, in November of 2002, the groups took him a notebook full of those candidates.

“He did not initiate our project,” Levin says. “He was the recipient of a project we put together.”

Levin allows, however, that Romney was a “willing participant with us.” She says Romney designated Kerry Healey — now a top Romney adviser — as liaison to the groups.

At first, their collaboration brought results. “He did increase the percentage of women who were in state government, which we appreciated,” Levin says. However, by the end of his administration, she adds, the level of women in government had slid back down again.

Despite the flare-up, Romney's running mate Paul Ryan defended his "binders" comment, and told Matt Laurer on the Today show he thought Romney "won the debate." Watch his interview this morning below.

Visit NBCNews.com for breaking news, world news, and news about the economy

Romney also took some heat for his stance on the Lilly Ledbetter Fair Pay Act, but top-tier supporters and former associates like Kerry Healey are backing the candidate.

From Talking Points Memo:

“I just want to point out that when Governor Romney’s campaign was asked about the Lilly Ledbetter bill, whether he supported it, he said, ‘I’ll get back to you,’” Obama said in the debate. “And that’s not the kind of advocacy that women need in any economy.”

TPM asked Kerry Healey, Romney’s former lieutenant governor in Massachusetts, if she could rebut Obama by describing the governor’s position on that bill.

“He’s not going to have his campaign defined by what the Obama campaign wants to talk about,” she replied. “Governor Romney gave the answer that women need to know, which is that he has deliberately gone out and made sure that women were well represented and well treated and respected in his workplace and he wants to make sure that that’s the case in workplaces around the country.”

As for the specific bill: “Saying ‘will you sign this, would you support that,’ this is just a campaign tactic,” she said. “I think you should look at what Mitt Romney has done.”

A USA TODAY/Gallup poll showed Romney leading with women voters in swing states before the debate, during a period between Oct. 5-11. But this development may have swung things back in Obama's favor.

With the second matchup behind them, Romney and Obama headed back out onto the campaign trail Wednesday. The president is heading to Iowa and Ohio, while Romney takes on another swing state, Virginia. 

Read more of Neon Tommy's debate coverage here.

Reach Executive Producer Catherine Green here. Follow her here.