The state-run Anadolu Agency reported yesterday that two members of the right-wing group, Yalcin Semir Akarsu and Cenk Kizilirmak, were placed under house arrest after the incident that occurred Saturday at the Incirlik air base in Adana province.
There was no immediate reaction from US or Turkish officials. Turkey's private Dogan news agency says Kizilirmak filmed his friend as he chased the soldier.
The nationalist Turkish Youth Union posted footage on Twitter in which the man identified by Dogan as Akarsu is heard as saying, "You put a sack over our soldiers' heads in 2003. You are responsible for that and for the terrorism in our country."
They were released after 60 hours in response to protests by top Turkish leaders, but what became known as "the Hood event" was seen by many Turks as a deliberate insult.
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Akarsu also was referring to the recent series of bombings in Turkey since last summer, some claimed by Kurdish militants and others blamed by the authorities on the Islamic State group.
"You cannot leave your base, but we can come to your prison and put this sack (on your head)," he said.
Last month, the State Department and Pentagon ordered the families of US diplomats and military personnel Tuesday to leave posts in southern Turkey due to "increased threats from terrorist groups" in the country.