Carolyn Kaster / AP A woman in the audience stands as Sarah Palin speaks during the 40th annual Conservative Political Action Conference in National Harbor, Md., March 16, 2013.

If you want to understand the biggest challenge facing the Republican Party, there are worse places to start than a private mailbox located near an Applebee’s in a Nashua, N.H., shopping center. This is the official address of the Hillary Project, a new super PAC dedicated to blocking Hillary Clinton from becoming President. This obscure political startup, whose website catalogs the various sins of the former Senator and Secretary of State — “the name alone strikes dread in the hearts of freedom-loving Americans” — might have escaped notice had the proprietors not included a charming game in which Web visitors are encouraged to slap a pink-pantsuit-clad Clinton with a cartoon hand.

You could write this off as some loser’s sick, sexist joke. But there was also the popular evangelical blog that depicted Clinton in ghoulish Joker makeup and inveighed against the danger of electing a female President who “wouldn’t be taken seriously.” Or the conservatives who dubbed Democratic state senator Wendy Davis “Abortion Barbie” after Davis filibustered a Texas bill that curbed abortion rights, in an apparent dig at her appearance.

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These isolated incidents don’t represent the values of the Republican Party, but taken together they may encapsulate an existential threat to it. Barack Obama was re-elected because he won women by 11 points. It was the fifth presidential election in a row in which Democrats won a majority of female voters, who make up 53% of the electorate. And while Republicans don’t need to eliminate the gender gap to recapture the White House, they will have a hard time doing so if the deficit increases. With the specter of a historic Clinton candidacy looming in 2016, Republicans are facing the daunting task of defeating the first woman with a shot at becoming President without alienating female voters.

“It doesn’t take a rocket scientist to see there was a problem with women in the last election cycle,” says Sharon Day, co-chair of the Republican National Committee. “I’m not delusional — I understand we have a lot of work to do.”

The RNC has launched a wide-ranging effort to attract more female candidates. “Our biggest problem was a lack of women representing us,” Day says. The party aims to remedy that by providing training and tactical support to prospective female candidates at all levels of government, as well as ramping up its recruitment efforts. That includes a new video in which female Republicans in Congress urge more women to join them; the new RNC Rising Stars program designed to showcase diversity within the party’s ranks; and doubling down on efforts to highlight elements of the Republican platform, from fiscal conservatism to school choice, that may hold appeal.

Day acknowledged the “Slap Hillary” game was “inappropriate.” But while Republican officials recognize the problem, there is a segment of the party that seems bent on making it worse. From calling Davis “Retard Barbie” to Rush Limbaugh’s ugly diatribe against Sandra Fluke to the Fox panels in which a quartet of mostly conservative dudes bemoan the rising trend in female breadwinners, there is a market within the GOP for misogynistic rhetoric. Every time the Erick Ericksons of the conservative movement sound off about the male’s rightful “dominant role,” or an elected official bumbles, Todd Akin–style, into a statement about rape, it becomes harder for the GOP establishment to combat the cheap but effective messaging trope that the GOP is engaged in a “war on women.”

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Neither party, of course, has a monopoly on misogynists or boors. Opponents of Sarah Palin started their own website that challenged browsers to slap the former governor “back to Alaska”; porn producers put a Palin lookalike in X-rated films. The Anthony Weiner–Eliot Spitzer–Bob Filner triumvirate is not a winning advertisement for male enlightenment within the Democratic Party.

But embarrassing episodes can be especially damaging to Republicans, whose values on abortion can make it tough to court women who cherish their right to the practice. While the percentage of Americans who call themselves “pro-choice” has dwindled, a majority still believe abortion should be legal in most or all cases. “Candidates have to be able to discuss this issue, and other so-called women’s issues, in a way that is modern and contemporary rather than sounding like the party wants to return to the 1950s,” says Liz Mair, a pro-choice Republican consultant who has been outspoken about the challenges the party faces in broadening its appeal.

“I think after last year’s election, party leaders have a pretty good grasp of exactly how big a problem the Akin comments, Santorum-style rhetoric” and the like can be, Mair says. But “there is only so much party leaders can do at the end of the day — just ask the Democrats in reference to the Filner fiasco, Anthony Weiner running for NYC mayor, or Alvin Greene’s nomination.”

For its part, the RNC says it is committed to winning back women, largely by recruiting a new crop of candidates who can be effective messengers. “We want our party to look like America,” Day says. “We need women to step up and lead.” But when asked about the timetable to close the gender gap, she doesn’t mince words: “It’s going to be a while.”

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