Sensing the challenge, the mainstream parties have been quick to defend their records. Kate Green, Labour’s shadow Minister for Women and Equalities, said in an email that it was the Labour Party that extended maternity leave to a full year and that introduced the Equal Pay Act, the Sex Discrimination Act and the Equality Act.

In July, Prime Minister David Cameron, of the Conservative Party, announced that he would introduce rules by the middle of next year to make companies with more than 250 employees publish the differences between the average pay of their male workers and that of their female employees.

For WE, those steps have not been enough. Its goals include having women make up 66 percent of the candidates to replace retiring members of Parliament and 75 percent of new peers appointed to the House of Lords. The high percentages are needed, they say, to quickly correct the lopsided representation of men.

Ms. Mayer said she never seriously considered joining one of Britain’s mainstream parties and lobbying for women’s equality from within because “the one way you can very quickly change the minds of mainstream parties is threatening them at the ballot box.”

In a separate interview, Ms. Walker agreed. “When there’s a political risk for mainstream parties,” they begin to listen and change policies, she said. “We’ve seen this happen with UKIP, the Green Party and the Scottish National Party.”

Though she disagrees with the policies of the U.K. Independence Party, or UKIP, Ms. Toksvig says it is the “template.” While the party won only one seat in the May general election, she said, “they managed to put a European referendum on the table.”

Others are skeptical. Tim Bale, a professor of political science at Queen Mary University of London, doubts that the Women’s Equality Party will command the same level of support.