“Everyone has a common agreement there’s a major problem in our immigration system, but it’s going to take leadership, from the president and from the Senate,” said Representative James Lankford, Republican of Oklahoma, who is running for an open Senate seat in his home state. “The conversation kept circling around to, ‘How can we possibly trust this president?’ ”

Still, the conservative groups, sensing an opening, clearly provided the arguments that prevailed. Heritage Foundation figures like Robert Rector provided intellectual heft for Republicans looking for arguments to oppose their leaders. As House members gathered in Cambridge, Md., to hash over the issue, Heritage posted on its Foundry blog the argument that would ultimately win the day.

Derrick Morgan, the think tank’s vice president for domestic and economic policy, was succinct and directive: “During their retreat, Republicans in Congress are considering what they should do. Congress should not trust the president as a partner to push an unpopular amnesty.”

For good measure, Heritage posted a video featuring Mr. Morgan making that case — and opening with a call for action on immigration from Mr. Obama, a messenger most conservatives reflexively resist. Invitations had already been delivered for what Heritage Action was billing as an alternative policy retreat, to give lawmakers opposing their own leadership new objectives to embrace.

And in the days that followed the retreat, such efforts continued. On Wednesday, just a day before Mr. Boehner all but pulled the plug, Representative Steve King, Republican of Iowa, one of the most committed opponents of immigration legislation, huddled with a small group of conservative members and the pollster Scott Rasmussen. Mr. King said that Mr. Rasmussen’s message was pointed: “I can’t think of a stupider thing for your conference to take up,” he said. (One week earlier at a similar gathering, Mr. King added, the guest speaker had been Sean Hannity of Fox News, who had sounded a similar alarm.)

That same day, Mr. Bozell’s organization, ForAmerica, took to Facebook to urge its members to call Mr. Boehner with a simple but urgent message. “Border security should be stressed and should be the priority first and foremost,” said David Bozell, the group’s executive director and the son of L. Brent Bozell. “And until that is accomplished, we really don’t trust any branch of the government to put together a package.” The younger Mr. Bozell said that the organization made over 5,500 calls. “We certainly think there was an impact there,” he said.

Behind the scenes, Representative Trey Gowdy, Republican of South Carolina and chairman of the House Judiciary Subcommittee on Immigration, began talking about the importance of dealing with the nation’s immigration problem — but not until 2015, a critical shift in strategy.