WASHINGTON — The Republican establishment’s much-anticipated pushback against the tea party wing is underway. House Speaker John Boehner made that clear Thursday, when he renewed his denunciation of groups that try to defeat GOP incumbents they consider too willing to compromise with Democrats.

Some Republican loyalists wonder what took so long. The U.S. Chamber of Commerce recently took steps to help mainstream Republicans in party primaries, but Boehner’s high-profile outburst will move the effort to the GOP’s front burners.

Cheering him on are mainstream Republicans who angrily watched for three years as hard-right groups exercised remarkable clout in the party, the Congress and elections. Tea party-backed nominees helped the GOP win control of the House in 2010, but they also lost several Senate races seen as winnable, keeping the Senate in Democratic hands.

This past summer, uncompromising House Republicans forced a partial government shutdown that damaged the party’s image, just as Boehner warned it would.

Many Republicans also feel conservative activists pushed presidential nominee Mitt Romney so far to the right on immigration and other issues that it eased President Barack Obama’s path to re-election last year.

“The establishment has no choice at this point,” said former Rep. Tom Davis, a Virginia Republican who has criticized the tea party’s growing influence. “So they’re taking them on.”

“To follow these groups is a downward spiral,” Davis said.

Conservative activists say Boehner and others will regret their moves. Groups like Heritage Action and the Club for Growth have tens of thousands of followers, said Rep. Tim Huelskamp, R-Kan., who has often feuded with Boehner.

“For any Republican to ignore them is dangerous to them electorally,” Huelskamp told reporters. Tea party activists were crucial to the Republicans’ breakthrough victories in 2010, he said. If party elders say, “We don’t need you anymore,” he said, millions of conservatives may sit out future elections.

Heritage Action spokesman Dan Holler said his group won’t back down. When Boehner writes off the dozens of House members who won’t compromise on tax and spending issues, it means “he’s going to rely heavily on Democrats” to pass legislation, Holler said. That’s bad for conservative principles, he said, and bad for GOP cohesion in elections.

Boehner’s allies say the alternative is worse. When Boehner tries to placate the staunchest conservatives in his caucus, they say, the results are a government shutdown, a major loss on the “fiscal cliff” deal a year ago and other Republican embarrassments.

Sixty-two House Republicans voted against the Boehner-backed budget deal Thursday, requiring dozens of Democratic votes to pass it.

Steve LaTourette, a Boehner friend and former GOP House member from Ohio, said he is heartened by the stepped-up actions by Boehner, the Chamber and others frustrated by tea party tactics.

He warned, however, that mainstream Republicans won’t tame the tea party faction without huge amounts of effort and money.

The intraparty struggle, LaTourette said, “is a script that’s yet to be written.”