WASHINGTON (AP) — Midterm elections that will decide control of the Senate are three months away, and the 2016 presidential campaign will start in earnest soon after. Yet the Republican Party still can’t figure out what to do about illegal immigration.

It’s the issue that vexed Republicans as much as any in their 2012 presidential loss. It’s the one problem the party declared it must resolve to win future presidential races. And it still managed to bedevil the party again last week, when House Republicans splintered and stumbled for a day before passing a face-saving bill late Friday night.

The fiasco proved anew that a small number of uncompromising conservatives have the power to hamper the efforts of GOP leaders to craft coherent positions on key issues — including one that nearly two-thirds of Americans say is an important to them personally, according to an Associated Press-GfK poll released last week.

“It would be very bad for Republicans in the House not to offer their vision of how they would fix the problem,” South Carolina GOP Sen. Lindsey Graham said when the initial House bill on immigration collapsed. While Republicans in the House are able to reject the proposals of Democrats, Graham said, that’s not enough: “At least they have a vision.”

While often a flashpoint issue among Republicans in their primaries this year, the party could get a grace period of sorts in November. Immigration appears likely to have only a modest impact on the roughly 10 Senate races that will determine control of the chamber. The possible exception is the race between Democratic Sen. Mark Udall and GOP Rep. Cory Gardner in Colorado, where Hispanic voters made up 14 percent of the electorate in 2012.

Even if President Barack Obama moves ahead with a proposal to give work permits to millions of immigrants living in the country illegally, removing the threat of deportation, Democratic strategists say Republicans won’t reap much of a benefit. Republicans, they argue, have already squeezed as much as they can from voters angry at the president by hammering at his record on health care, the IRS, foreign policy and other issues.

“There’s a ceiling, and nothing the president can do can get them above the ceiling,” said Rep. Steve Israel of New York, head of the Democrats’ efforts to win House elections. “But swing voters and persuadable voters, they want solutions.”

Hispanics made up less than 3 percent of all registered voters in 2012 in seven other states with competitive Senate races: Louisiana, Arkansas, North Carolina, Iowa, Michigan, Georgia and Kentucky. So any Democratic benefits from an Obama executive action on immigration could be just as limited.

Still, a few Democratic senators in those tight contests — including Kay Hagan of North Carolina and Mark Pryor of Arkansas — are putting some distance between themselves and the president. The White House, Pryor said, is “sending mixed messages: telling folks not to cross the border illegally and then turning around to hand out work permits to people who are already here illegally.”

Both parties agree that immigration is likely to play a bigger role in the 2016 presidential election. Arizona Sen. John McCain, the GOP nominee in 2008, has said his party can’t win without supporting an overhaul of the nation’s immigration laws, while former Florida Gov. Jeb Bush is among the potential candidates to urge the party to liberalize its approach to immigration.

A GOP-sanctioned “autopsy” of Mitt Romney’s 2012 loss made only one policy recommendation: The party “must embrace and champion comprehensive immigration reform,” a term understood to include creating pathways to legal status for millions of immigrants living in the nation illegally.

For that reason, some Republicans found the House hubbub discouraging. Party leaders had to yank an immigration bill from the floor Thursday after realizing they lacked the votes to pass it. Democrats mocked House Speaker John Boehner for declaring that Obama should take numerous steps, “right now, without the need for congressional action, to secure our borders,” while his website also stated, “More unilateral action from the White House will make (the) border crisis worse.”

“I’m just about as conservative, and full-spectrum conservative as it gets, and I was going to go yes” on Thursday, said Arizona GOP Rep. Trent Franks. “So I’m not certain what happened.”

Ultimately, the party’s rank-and-file refused to start Congress’ five-week break without proving the GOP could pass some type of immigration bill. It would clear the way for eventual deportation of more than 700,000 immigrants brought here illegally as children. It also would allocate $694 million for border security efforts, including $35 million for the National Guard.

The action kept Republicans from ending the summer empty-handed on immigration. But that doesn’t mean the party is any closer to untying the nation’s immigration knot.

While solid majorities of Americans say the country’s current immigration policies are unacceptable, many House Republicans owe their jobs to conservative activists who fiercely oppose “amnesty” for immigrants and dominate GOP primaries in districts where Democrats have almost no chance of winning.

Some of those Republicans were among the House conservatives who met last week in the office of Texas Republican Sen. Ted Cruz, who urged them to force concessions from Boehner’s leadership team. And on Friday, Cruz was talking about immigration in the Senate race in New Hampshire, which will hold the first presidential primary of 2016.

In a fundraising message, Cruz attacked Democratic Sen. Jeanne Shaheen for supporting Obama’s “amnesty” immigration policies.

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Associated Press writers Bill Barrow in Georgia, Thomas Beaumont in Iowa, Steve Peoples in New Hampshire and Jim Kuhnhenn at the White House contributed to this report.

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