Acknowledging the party’s mistakes with female voters, Republicans fielded female candidates in the 2014 midterms who could appeal more broadly. These candidates played down their social conservatism and emphasized their biographies as glass-breaking female leaders. In 2014, Martha McSally, the nation’s first female fighter pilot to serve in combat, won the Republican primary in a swing district in Arizona. Elise Stefanik, then the youngest congresswoman ever elected, was chosen to be co-chairwoman of the party’s moderate caucus.

With the exception of her endorsement of Senator Joni Ernst of Iowa, Ms. Palin played a minor role in the 2014 midterm victories. Her days as queen-maker seemed over. Most signs suggested that Republican pragmatism would prevail going forward.

The rise of Donald Trump eliminated any chance of that. The signs of women’s disillusionment with the party were immediate. One day after President Trump’s inauguration, an estimated four million people, mostly women, participated in hundreds of women’s marches throughout the United States. In the months that followed, women mobilized to defeat Republicans at the federal, state and local levels.

In the 2018 midterms, every Republican congresswoman from the 2014 class except for Ms. Stefanik lost. Martha McSally was defeated, though she was appointed by Arizona’s Republican governor to fill John McCain’s Senate seat on his death. She faces a tough election battle in 2020.

Despite winning three governors’ races and one open Senate seat in 2018, Republican women will end the decade with their governors and senators outnumbered two to one by their Democratic counterparts. Three of the four current female Republican senators running in 2020 face highly competitive elections in 2020. There are more than six times as many Democratic women as Republican women in the House.

Granted, Republicans — both men and women — have suffered dramatic losses in elections at national, state and local levels since 2016; they have lost control of the House, eight governorships and nine state legislative chambers since Mr. Trump’s election. But female Republican candidates and officeholders face an existential threat.

Mr. Trump’s misogyny and the party’s far-right stance on issues such as abortion and L.G.B.T.Q. rights, guns and immigration have driven away many female voters. Women favor the Democratic Party over the Republican Party by a 19-point margin, according to the Pew Research Center. Seventy-three percent of women under the age of 30 disapprove of the president’s performance, according to the Harvard Institute of Politics Youth Poll.