Within the orbit of Muslim-American activists, Ms. Ahmed has earned a combination of head-scratching and credibility. Although she says the Republican Muslim Coalition has a $2.5 million annual budget and intends to start local chapters, it appears very much to be a one-woman show. Yet it is also clear that she has rapidly made herself known in influential corners of the conservative universe.

“It’s actually very healthy for some Muslims to remain in the Republican Party, because they act as a counterweight to Islamophobia,” said Robert McCaw, manager of the government-affairs department at the Council of American-Islamic Relations, a civil rights group. “And I think she’s useful for the party. While they benefit from fringe voters’ animosity toward Muslims, they don’t want to be publicly perceived by the majority of Americans as being racist and exclusionary.”

Khurrum Wahid, the founder and chairman of Emerge USA, an American Muslim lobby active in swing-state elections, made a similar point: “Without Muslims at the Republican Party table, Muslims who identify with the issues of the broader American Muslim community, there can be no progress nationally on issues such as Islamophobia.”