House Republican leaders will face a familiar dilemma this week when they try again to approve funding to keep the Department of Homeland Security functioning through the end of September: They know their party is too divided to resolve the crisis on its own but fear the political fallout if they rely on Democrats to get them out of the jam.

After a humiliating defeat of their original funding plan on Friday afternoon, House Speaker John Boehner, R-Ohio, has just five days to craft a new outline to keep DHS funded or face the politically debilitating prospect of at least temporarily shutting down an agency designed to protect Americans.

By late Sunday, Boehner’s House Republicans had no clear path to a solution other than retreating from their demands that the DHS funding measure include provisions that would block implementation of President Obama’s executive actions on immigration.

A bill without the immigration language would be likely to pass because it would attract a huge bloc of Democratic support. But it would bitterly divide Boehner’s frustrated caucus.

That prospect left Boehner facing questions on Sunday about whether he even enjoys his job anymore. “On most days,” the speaker said Sunday on CBS’s “Face the Nation.” “Friday wasn’t a whole lot of fun. But most days.”

Boehner and his top lieutenants fanned out across Sunday’s talk shows, seeking to portray the Friday defeat of their original funding bill – which would have extended funding authority for the DHS for three more weeks – as a dispute over “tactics,” about how to confront the president on his immigration actions, which they regard as an abuse of his power. They dismiss the notion that the bill’s failure signaled a more profound divide between far-right conservative and establishment GOP factions.

“We have a difference of opinion in strategy and tactics, but in principle we are united,” House Majority Leader Kevin McCarthy (R-Calif.) said on NBC’s “Meet the Press.” “We are united in the principle there’s a right way and wrong way to legislate. Unfortunately, the president chose the wrong way.”

But the factions have broken open into very emotional divisions.

On one side is a majority of conservatives, who support the leadership’s tactics and believe in trying to govern.

Over the weekend, one of Boehner’s closest allies, Rep. Devin Nunes (R-Calif.), issued a statement that blamed the leadership’s troubles on “a small group of phony conservative members who have no credible policy proposals and no political strategy to stop Obama’s lawlessness.”

That small group is led by a breakaway faction of Republicans working under the banner of the Freedom Caucus, 10 lawmakers who repeatedly object to leadership’s moves as insufficiently conservative.

This group decided it no longer trusted the original conservative caucus, the Republican Study Committee, and formed their own smaller caucus.

Closely allied with outside conservative groups who have agitated for an aggressive line, even if it means shutting down the DHS, that caucus led the way in rallying 52 Republicans to oppose Boehner’s three-week funding plan and, initially without Democratic support, tanked that vote.