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The U.S. government believes some Americans have joined the Islamic State and, programmed to conduct suicide bombings, will pose risks upon their return.

Britain says hundreds of its nationals have joined the militants and have committed some of its most brutal killings. British authorities said Wednesday that they suspect the black-clad executioner in Tuesday’s beheading video is a Briton.

Prime Minister David Cameron said the man had not been identified, but “from what we have seen it looks increasingly likely that is a British citizen.”

Germany and France say about 1,300 of their nationals have joined jihadist groups, including the Islamic State, The Associated Press reported.

In the U.S., the Islamic State flag has appeared sporadically as the terrorist group’s propaganda arm tries to recruit young killers.

“They have said they are going to go after Americans,” Mr. Spencer said. “They have said American civilians are all legitimate targets. It seems to be absolutely certain they will be trying to kill as many Americans as possible in the United States as well as elsewhere.”

Retired Army Gen. John M. Keane, who is pressing the White House to adopt a broad campaign against the Islamic State, agrees that the terrorist organization of more than 10,000 fighters represents wider threat.

“[The Islamic State] is the new face of al Qaeda and is a threat to U.S. vital interests in the Middle East and a threat to the safety and security of the American people,” Mr. Keane said. “Despite some limited setbacks in northern Iraq, IS maintains the initiative and freedom of movement throughout Iraq and Syria. What the U.S. needs is not just rhetoric condemning this barbarism but a U.S.-led strategy with our allies to first stop IS and then defeat them.”

The Islamic State has perfected the car bomb as a weapon. After U.S. troops left Iraq in 2011, it unleashed waves of suicide and remotely detonated car bombs throughout Baghdad and other cities.

A Canadian member of the Islamic State was quoted as saying he is training others on how to build such deadly weapons — a hint that such vehicle-borne improvised explosive devices may show up in his home country.

Retired Army Lt. Gen. William Boykin, formerly the Pentagon’s No. 2 intelligence official, said the Islamic State’s game plan is to threaten the U.S. repeatedly to raise its stature among young potential jihadists.

He said the terrorist group also believes it can intimidate Mr. Obama into halting the roughly 75 airstrikes that have helped Iraqi Kurds protect their capital of Irbil and retake the Mosul Dam.

“The notion is that Barack Obama is weak right now and he will fall back on appeasing his base, which is very much against any continuation of military ops in Iraq,” Gen. Boykin said. “I personally believe that this notion is a miscalculation by IS. Obama is under too much pressure now from inside his own party to try to convince anyone that the U.S. should declare victory and go home.”

The Obama administration has sent mixed messages on the threat posed by the Islamic State, which previously called itself the Islamic State in Iraq and the Levant.

In January, Mr. Obama seemed to brush off any dangers, comparing it in an interview with The New Yorker to a “JV team” compared with al Qaeda.

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