WASHINGTON — Fresh off their big midterm elections victory, congressional Republicans thought they might be on the offensive against a chastened President Barack Obama right about now.

Instead, Obama’s executive order on immigration has them scrambling for a response that satisfies irate conservatives without alienating moderates, Hispanics and other voters who will be crucial for the 2016 presidential election.

The situation poses a major challenge for House Speaker John Boehner, R-Ohio, and incoming Senate Majority Leader Mitch McConnell, R-Ky., as they inaugurate their newly minted congressional majority on a familiar note of GOP infighting that party leaders sought to avoid. Boehner scheduled a news conference for Friday morning to respond to Obama’s immigration plans.

Whether the GOP leaders are able to rein in the tea party contingent in Congress that forced a politically damaging government shutdown a year ago over the president’s health care law is an open question. The answer will have major implications for whether the GOP can hang onto its newfound control of Congress, and hope to win the White House in two years.

“What did the president do? He pulled the pin on the grenade two weeks after the election,” said Rep. Pat Tiberi, R-Ohio, a close Boehner ally. “I don’t think anybody knows or can predict what happens and the carnage that this creates quite frankly for the legislative process.”

On one point, Republicans are united: Obama must be stopped.

“It is our obligation and responsibility to fight this brazen power grab that doesn’t solve the real problems,” House Majority Leader Kevin McCarthy, R-Calif., said following the president’s announcement of an executive order to shield as many as 5 million immigrants here illegally from deportation, and grant them work permits. “Our Constitution does not grant the president the authority to legalize millions of immigrants with the stroke of a pen.”

But how, exactly, Congress could stop Obama remained far from clear.

Obama’s announcement presented party leaders with a knotty dilemma: How to undo an action set to take effect without Congress doing anything, with no obvious legislative vehicle for doing so and Obama able to veto any legislative solution they derive.

Conservative lawmakers are pushing to insert language in upcoming must-pass spending bills to block Obama’s order. Party leaders warn that could lead to a government shutdown.

Numerous Republicans have discussed suing the president over his immigration orders, or expanding a lawsuit already planned over the health care law to include immigration. Yet they fret it would take too long and wouldn’t have the effect of blocking Obama’s orders from going into effect.

Others said that House Republicans, who never acted on a comprehensive immigration bill passed last year by the Senate, had an obligation to try to craft their own legislation. But chances of success seemed remote, at best.

As Democrats largely cheered Obama’s announcement and the prospect of an enthusiasm boost among Latino voters who make up an increasingly large share of the electorate, GOP debate flared between establishment Republicans and the tea party wing.

House Appropriations Chairman Hal Rogers, R-Ky., declared that it was impossible to “defund” the U.S. Citizenship and Immigration Services, since it pays for itself based on application fees. He’s pushing for passing a yearlong spending bill to get spending fights out of the way, and then finding some other way to respond to Obama.

That angered some conservatives who argued that establishment Republicans were just looking for a way out of a necessary confrontation with the president.

“They’re contriving red herring arguments to get to the point where enough members will walk out of this Congress and go home for Thanksgiving and say, ”˜Well, there’s nothing we can do,’” said Rep. Steve King, R-Iowa, who was among a small group of conservatives arguing impeachment should be on the table as a last resort. “I feel sorry for them if they step up in front of their constituents and say ”˜Congress can’t do anything, we have to let the president violate the Constitution.’ “

Republican lawmakers reported being deluged by calls and emails from constituents demanding they stand up to Obama. Publicly and privately, party leaders urged restraint, but there were concerns that on a combustible issue like immigration, restraint would be difficult.

“It throws the place in a bit of upheaval,” said Sen. Lindsey Graham, R-S.C. “Our goal is to not capitulate, but not overreact by pushing impeachment or shutting down the government.”