Here's the kicker though: the Turkmen Syrian group is led by an alleged member of the Grey Wolves organization
In the late 1970s, former military prosecutor and Turkish Supreme Court Justice Emin Değer documented collaboration between the Grey Wolves and Counter-Guerrilla—the Turkish stay-behind anti-communist organization, part of NATO planning which was supposed to prepare networks for guerrilla warfare in case of a Soviet invasion—and the Central Intelligence Agency (CIA). Martin Lee writes that the Counter-Guerrilla supplied weapons to the Grey Wolves, while according to Tim Jacoby, the CIA overtly transferred guns and explosives to Grey Wolf units through an agent in the 1970s.
During the Susurluk scandal of 1996 the Grey Wolves were accused of being members of the Counter-Guerrilla, the Turkish branch of Operation Gladio. Abdullah Çatlı, second in command of the Grey Wolves leadership, was killed during the Susurluk car crash, which sparked the scandal. The April 1997 report of the Turkish National Assembly's investigative committee "offered considerable evidence of close ties between state authorities and criminal gangs, including the use of the Grey Wolves to carry out illegal activities."
In the 2008 the Ergenekon trials a court document revealed that the National Intelligence Organization (MİT) armed and funded Grey Wolves members to carry out political murders. They mostly targeted members of the Armenian Secret Army for the Liberation of Armenia (ASALA), which attacked Turkish embassies abroad in retaliation of the denial of the Armenian Genocide. The Turkish intelligence services also made use of the Grey Wolves in the conflict against the Kurdistan Workers' Party (PKK) by offering them amnesty in exchange.
Their helicopter was shot down by an American weapon, given to a man belonging to a group like that? The GRU are not strong believers in coincidence. They think the provocation was staged (and not without good reason), so they're almost certainly considering the followup in a similar light.
Personally, I do think it was coincidence, at least with regards to the weapon. Lots of American weapons in that part of the world, they could've picked it up anywhere. But it does pay to be suspicious of Turkey.