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As Iowa Republicans headed to the caucuses on Monday night, Senator Ted Cruz’s campaign left recorded messages for supporters with “breaking news” that Ben Carson would drop out of the race, and told them to “inform any Carson caucus-goers of this news and urge them to caucus for Ted instead.”

The false report, echoed in an email and in a text message sent to campaign volunteers, was trumpeted by at least some Cruz precinct captains when they addressed their caucuses. When Mr. Carson’s wife, Candy Carson, arrived at two precincts to speak on his behalf, she was furious to learn that speakers for Mr. Cruz had suggested moments earlier that her husband was quitting the race.

The Cruz campaign on Friday acknowledged it had made a coordinated effort to spread the story. But it defended its actions as an honest mistake based on “reports,” namely CNN anchors echoing Twitter messages from a reporter saying that Mr. Carson was heading home to Florida after Iowa, rather than to New Hampshire or South Carolina, where the next contests were to be held. However, those messages were followed almost instantly by another from one of the reporters stating that Mr. Carson would remain in the race “no matter what.” A senior strategist for Mr. Carson, Jason Osborne, had reiterated on Twitter: “Not standing down.”

The Carson campaign, which has angrily accused Mr. Cruz of dirty tricks, escalated the feud on Friday by using the audio recording of the message left by Cruz supporters in a fund-raising email. “Hello,” the call began, “this is the Cruz campaign with breaking news: Dr. Ben Carson will be suspending campaigning following tonight’s caucuses.”

Mr. Cruz, who won the Republican caucuses, apologized to Mr. Carson this week. At a news conference in Washington, Mr. Carson said that Mr. Cruz had not gone far enough in addressing the situation and called on him to fire the staff members who spread the false rumors.

Mr. Carson’s fourth-place showing in Iowa, where he got 9.3 percent of the vote, was equal to or slightly better than his support in polls before the caucuses, raising doubts about whether the Cruz disinformation swayed many voters. Nonetheless, the issue has become a distraction to Mr. Carson ahead of the New Hampshire primary on Tuesday, and it has raised questions about the tactics of the Cruz campaign.

“It’s really demoralizing. People are angry,” Mr. Osborne said. “Every day, as more information comes out, he’s getting more animated about it,” he added, referring to Mr. Carson.

Since Mr. Carson rose to the top of Iowa polls last year, his campaign has faced major upheavals. This week he shed staff members to cut costs. He had refrained from attacking his rivals throughout the campaign, but he has latched on to the Cruz tactics to garner attention and, now, to raise money.

A spokeswoman for Mr. Cruz defended the voice mail messages, saying they were not deceptive because they had said Mr. Carson was “suspending campaigning” by not going directly to New Hampshire from Iowa.

“Our campaign shared an accurate report that Carson was suspending campaigning after the caucuses — he went home and he went to D.C. — and these voice mails do not suggest that he would completely drop out of the race,” said the spokeswoman, Catherine Frazier.

While perhaps narrowly correct, it is largely understood that suspending a campaign means a candidate is quitting. That was how Representative Steve King, Republican of Iowa, who endorsed Mr. Cruz, interpreted the report in a Twitter message he posted about 20 minutes after the caucuses began on Monday: “Skipping NH & SC the equivalent of suspending. Too bad this information won’t get to all caucus goers.”

A minute later, Mr. King added: “Carson looks like he is out.”

The Carson campaign prepared a timeline for its internal use showing that the episode began at 6:43 p.m. in Iowa when a CNN reporter, Chris Moody, said on Twitter, “Carson won’t go to NH/SC, but instead will head home to Florida for some R&R.”

Moments later Mr. Moody tweeted, “Ben Carson’s campaign tells me he plans to stay in the race beyond Iowa no matter what the results are tonight.”

CNN anchors on a live broadcast almost simultaneously repeated Mr. Moody’s tweet that Mr. Carson was taking a break from the campaign trail, but not the campaign’s assertion he was not dropping out.

Ten minutes later, Mr. Osborne, who was traveling with Mr. Carson, clarified in a tweet that the candidate “will be going to Florida to get fresh clothes b4 heading out on the campaign trail.”

At 7:07 p.m., as caucuses across the state were being called to order, the first of two recorded calls went out to Cruz volunteers telling them to spread the word to Carson supporters that they should “not waste a vote for Ben Carson and vote for Ted Cruz.”

Mr. Cruz has largely dismissed questions about his campaign’s behavior on Monday night. Speaking to reporters on Thursday, before Mr. Carson released the audio, Mr. Cruz said it was “part of the democratic process to pass on things that are true and accurate.”

“I’m not sure y’all would want that to be considered unethical, passing on the stories that each of you write,” he said.

He made no distinction between a reported article and 140-character Twitter post echoed in a TV broadcast, one that provided the Cruz campaign with an opportunity to try to undermine Mr. Carson.

Matt Flegenheimer contributed reporting