“It’s one of them Facebook things where you see Muslim men are attacking women, and stuff like that, and having no respect for them at all. I’ve got a real issue with that,” he said. “I see story after story after story, and I don’t know how true it is.”

Mr. Eason said he was skeptical of stories by unfamiliar websites like American News, but he reads them anyway. He commented on one that was posted to Facebook:

Freedom’s Final Stand, Oct. 3, 2016: Muslim Refugee Beats and Rapes Woman in North Dakota While Yelling ‘Allahu Akbar’

The post, which was shared 14,000 times, linked to a story about a case in which the authorities have not described the immigration status of the suspect, or said that he was a noncitizen. They have also discredited the claim that the man yelled “Allahu akbar” during the episode.

Nevertheless, the comment that Mr. Eason posted on the site, which he later said he could not remember making, garnered 87 likes. “If Muslims are taught hate by their religion,” he wrote, “then all Muslims are potential terrorists and should be treated accordingly. TRUMP will stop this kind of stuff.”

Sgt. Timothy Briggeman of the Cass County Sheriff’s Office in North Dakota, which is investigating the case, said such stories and responses often appeared on social media when a person in his jurisdiction with an Arabic-sounding name is charged with a crime.

“To be honest,” he said, “it’s embarrassing and it’s disheartening when anyone with a name of such ends up in the news — the comments that get thrown around. That seems to be the No. 1 remark: ‘Send them back and get rid of them,’ and, ‘We don’t need them.’”

Worries that refugees might be radicalized have also been amplified on the internet. This story was shared at least 1,400 times:

American News, Jan. 11, 2016: Al Qaeda Terrorists That Killed Soldiers Have Entered the U.S. Disguised as Refugees

And this one, posted by The Daily Caller, was shared more than 3,000 times, despite linking to a story with no evidence of a cover-up:

The Daily Caller, Dec. 7, 2015: Homeland Security Chair: Obama Covering Up Evidence ISIS Is Targeting Refugee Plan

The actual number of refugees who have become extremists in the United States has been estimated at between three and 12 — out of the more than 800,000 who have resettled here since the Sept. 11, 2001, attacks.

The online stories about refugees range from outright fake news to those based on a grain of truth and then stretched out of proportion. For example, the Breitbart article about genital mutilation was based on a study that estimated that a half-million women currently living in the United States have had their genitals mutilated. But most of them were immigrants who had fled here because of such treatment in their home countries.