The tactic is making for some awkward moments online. The A.F.L.-C.I.O. sent to its more than 60,000 followers an America Rising tweet praising its president, Richard L. Trumka, for a speech that was seen as challenging Mrs. Clinton on economic issues, only to take it down a few hours later, saying it was a mistake.

Laura Hart Cole of Verbank, N.Y., whose father, Philip A. Hart, was a senator from Michigan and a liberal icon, was shocked to learn that she had, like Mr. McKibben, shared the meme from America Rising on Twitter. Republican groups, she said, “have a history of sleazy tactics.” But she added: “I guess it’s fair. If what they’re saying is factual, then I guess it’s fair play. It’s a dirty game.”

Conservative strategists and operatives say they are simply filling a vacuum on the far left, as well as applying the lesson they learned in 2012, when they watched in frustration as Mitt Romney was forced to expend time and resources in a protracted primary fight. By the time he secured his party’s nomination, President Obama hardly had to make the case that his opponent was a coldhearted plutocrat; Republicans like Newt Gingrich had already made the argument for him in the primaries.

Few Republicans are more familiar with that nightmare than Matt Rhoades, who was Mr. Romney’s campaign manager. He founded America Rising in response to a recommendation contained in an autopsy of Romney’s failed presidential run that was ordered by the Republican National Committee. The group’s original goal was to compete with American Bridge, the Democratic opposition research group, but its focus under Mr. Rhoades has been to subject Mrs. Clinton to an ordeal similar to Mr. Romney’s.

“The idea is to make her life difficult in the primary and challenge her from the left,” said Colin Reed, America Rising’s executive director. “We don’t want her to enter the general election not having been pushed from the left, so if we have opportunities — creative ways, especially online — to push her from the left, we’ll do it just to show those folks who she needs to turn out that she’s not in line with them.”