A New Firm Sets Out to Secure Women's Votes for a Vulnerable G.O.P. WASHINGTON - After months of deliberating over how to better appeal to Hispanic and other minority voters, some Republicans believe their party is overlooking another dire demographic challenge: women. One year after President Obama carried female voters by an 11 percentage point margin in his re-election, Republican officials are again grappling with another competitive race lost in large part because of women. In the Virginia governor's race last week, Terry McAuliffe defeated Kenneth T. Cuccinelli II, the state attorney general, by only 2.5 percent, but won by nine percentage points among women. The Republicans" challenge is even more acute with unmarried women: Mr. Obama won this often younger voting block by 36 percent. Mr. McAuliffe - who blitzed Mr. Cuccinelli with a multimillion-dollar barrage of negative ads on abortion, contraception and divorce - carried single women by a staggering 42 percentage points. "There were something like 53 million unmarried women eligible to vote in 2012, but on campaigns you don't hear a specific strategy discussed of "How are we going to reach unmarried women?" " said Katie Packer Gage, who served as deputy campaign manager for Mitt Romney in 2012. Given Democrats" repeated and successful efforts to portray Republicans as hostile to women, Ms. Gage and two other consultants are starting a political strategy firm to help the party's candidates better tailor their messages to women. Ms. Gage, Ashley O'Connor and Christine Matthews this week are opening what appears to be the first Republican firm aimed specially at wooing female voters. They are calling it Burning Glass Consulting - a reference to what they see as the need for a focus on appealing to women that is so laserlike that it can burn glass. "We want to get smarter about how we communicate the Republican message specifically to women," said Ms. Gage. Certainly there are challenges with other demographic groups, but women represent 53 percent of the electorate. The three strategists will undertake public opinion research, TV ads and general consulting for Republican candidates about how to better reach that majority. The idea for the firm came about when Ms. Gage and Ms. O'Connor worked together on the Romney campaign. The pair would find themselves among just a few women at meetings of about a dozen staff members. "We started to see things we felt weren't being accomplished," Ms. Gage said, citing Mr. Romney's message and tone. After the campaign they discussed their frustrations with Ms. Matthews, a pollster, and decided to create a new approach to both voter research and the campaign ads that are derived from such polling and focus groups. Ms. Matthews said that instead of commissioning a single poll or focus group among women, she was seeking "longer-term engagement" with a group of women, creating a relationship with them over a period of time to better assess what shapes their political views. Democrats dismiss such tactics, asserting that a consulting firm cannot change the core challenge, that Republicans have difficulties with female voters because of their party's policies and language. Democrats refer to the Republicans" approach as the "war on women." "The lesson for candidates in 2014 is unmistakable: Dismiss and demean women at your peril," Cecile Richards, the president of the Planned Parenthood Federation of America, said last week after the Virginia race. What is unmistakable is that Democrats think that in depicting Republicans as hostile to women, they have found a winning message. In Kentucky, home to perhaps the highest-profile Senate race next year, Democrats recruited Alison Lundergan Grimes, 34, the secretary of state, to run against Mitch McConnell, the Senate minority leader, and are already taking after him on gender-related issues. And, if Republicans nominate a man in Georgia, Democrats will surely do the same there, where they wooed Michelle Nunn, daughter of former Senator Sam Nunn, to run for an open Senate seat.