But according to local officials, both boys are Dutch. The clip was taken in Monnickendam, a small town in the North Holland province of the Netherlands, in May and shows a teenager punching and kicking another boy holding a crutch. Marleen van Fessem, a spokeswoman for the local public prosecutor’s office, confirmed the 16-year-old boy who was arrested after the video came to light was “born and raised in the Netherlands.”

The Embassy of the Netherlands in Washington then chided Mr. Trump. “Facts do matter,” it replied to the president on Twitter.

The two additional videos were taken in 2013, one in Syria, the other in Egypt, and are provided with no explanation of the political turmoil taking place in those countries at the time, and with no details on the extremist affiliations of one of the men in the video.

The president’s decision to share them was in keeping with his habit of disseminating information he has not verified even as he attacks news organizations for producing “fake news.” Two White House aides, who asked not to be identified because they were not authorized to speak publicly, said Mr. Trump found the videos himself. Aides said they believed he spotted one on the Twitter feed of Ann Coulter, the conservative commentator.

In an email later in the day, Ms. Coulter said that while she was still “annoyed” that Mr. Trump had not yet built the border wall he promised, “I LOVE the president’s tweets!” Responding to the British criticism, she said, “Maybe between blathering about the values of ‘tolerance and respect,’ poor Theresa May might want to ask herself whether the Muslims the U.K. is importing at breakneck speed share these ‘values of tolerance and respect.’”

Britain First was partly founded in 2011 by James Dowson, a far-right activist who left the group in 2014 and supported Mr. Trump during the 2016 presidential campaign. The organization calls itself a “patriotic” political party, but has been criticized by human rights activists as an extremist group that seeks to bait Muslims.