WASHINGTON — The right flank of John A. Boehner’s party may have pushed him out of the House speaker’s chair, but it will take members of every faction of the House Republicans to choose his successor. As the scramble to replace Mr. Boehner — and fill the leadership posts beneath him — begins in earnest this week, a few dozen members who spent several years tormenting the speaker feel deeply empowered in determining the outcome.

But while they may have effectively deposed Mr. Boehner, their own authority is in no way assured.

Representative Kevin McCarthy of California, who on Monday formally announced his effort to replace Mr. Boehner, has been plowing through his cellphone contact list to round up the votes that would make him the next speaker. But the calculus is complicated, with an equal number of self-described moderate Republicans and far-right members, as well as the vast numbers of those somewhere in between — roughly 150 other Republicans, who will be needed to reach the necessary 218 votes. Mr. McCarthy, who calls himself a conservative, appeals to some members of each group.

“We cannot have random government in which a small group hijacks the House,” said Representative Peter T. King, Republican of New York, who said that he would seek assurances from Mr. McCarthy and the candidates for majority leader and for the No. 3 post of majority whip that they will not be captive to a small group of members. “Whoever it happens to be needs to make reasonable attempts to reach all wings of the party.”

In a letter to his colleagues on Monday, Mr. McCarthy demonstrated what underlies the concerns of many members like Mr. King, taking veiled shots at Mr. Boehner and suggesting that as the next speaker he would take into account the faction whose rabble has been roused.