The struggles of House Republicans have been shown most recently in the emergence of an influential but unofficial group that could be called the Vote No/Hope Yes Caucus.

These are the small but significant number of Republican representatives who, on the recent legislation to head off the broad tax increases and spending cuts mandated by the so-called fiscal cliff, voted no while privately hoping — and at times even lobbying — in favor of the bill’s passage, given the potential harmful economic consequences otherwise.

Representative Tom Cole of Oklahoma, part of the Republican whip team responsible for marshaling support for legislation, said the current makeup of House Republicans could be divided roughly into a third who voted in favor of the bill because they wanted it to pass, a third who voted against the bill because they wanted it to fail, and a third who voted against the bill but had their fingers crossed that it would pass and avert a fiscal and political calamity.

One lawmaker, Mr. Cole said, told him that while he did not want to vote in favor of the bill, he also did not want to amend it and send it back to the Senate, where it might die and leave House Republicans blamed for tax increases. “So I said, ‘What you’re really telling me is that you want it to pass, but you don’t want to vote for it,’ ” recalled Mr. Cole, who voted yes.

The Vote No/Hope Yes group is perhaps the purest embodiment of the uneasy relationship between politics and pragmatism in the nation’s capital and a group whose very existence must be understood and dealt with as the Republican Party grapples with its future in the wake of the bruising 2012 elections.

Ron Bonjean, a Republican strategist and once the top spokesman for J. Dennis Hastert, a Republican and former House speaker, described the phenomenon thus: “These are people who are political realists, they’re political pragmatists who want to see progress made in Washington, but are politically constrained from making compromises because they will be challenged in the primary.”