At the core of the turmoil in the House Republican caucus is a group of about 40 members known as the “Freedom Caucus.” Supporters are cheering its recent impact and perceived clout, but others—including some in its ranks--say those actions could ultimately prove disastrous for the conservative cause its members champion.

The Freedom Caucus was formed in January, when a handful of lawmakers joined together to advance a more conservative agenda in the House. Their ranks, and ambitions, expanded from offering policy alternatives that would rein in government to one of their members, Mark Meadows of North Carolina, challengingJohn Boehner's speakership in July. After Mr. Boehner announced last month that he would resign as speaker, the Freedom Caucus endorsed Rep. Daniel Webster—preferring the Floridian hard-liner to House Majority Leader Kevin McCarthy of California. The caucus also demanded a greater role in the House leadership structure.

Mr. McCarthy withdrew from the contest to succeed Mr. Boehner partly, he has said, to avoid conservative primary challenges to members who supported his bid, the same reason Mr. Boehner cited.

When the Freedom Caucus began inserting itself into the leadership fight, Rep. Reid Ribble of Wisconsin decided to resign from the group. He made the decision shortly after Mr. Boehner announced he would step down, he told me on Thursday, but did not go public until Mr. McCarthy’s surprise withdrawal.

Mr. Ribble said he thought the Freedom Caucus was “pivoting away from real policy” to focus on “process.”