Or rather it might indicate that the leadership in both Turkey and Russia believe it to be true.
Whether or not it actually is true, isn't particularly important, and probably can't be proved anyway. The more important thing is what they believe to be true.
Considering the CIA had fingers in coups worldwide in the past means they have the capacity to organize or support 'regime changes'. In my country's case, it's known that the CIA had a role in the 12 September 1980 coup.
The American support of this coup was acknowledged by the CIA Ankara
station chief Paul Henze. After the government was overthrown, Henze cabled Washington, saying, "our boys [in Ankara] did it."
[32] Henze denied American involvement in the coup during a June 2003 interview on
CNN Türk's
Manşet, but two days later Birand presented an interview with Henze recorded in 1997 which confirmed Mehmet Ali Birand's account.
[33][34] The
US State Department itself announced the coup during the night between 11 and
12 September: the military had phoned the US embassy in
Ankara to alert them of the coup an hour in advance.
[9]
https://en.m.wikipedia.org/wiki/1980_Turkish_coup_d'état
The disappointed and unsupportive tone (over the failure of the 'coup' attempt by Gulenist) of the US and EU politicians and mainstream media is also very suspicious and alarming. All those overly negative propaganda (e.g. Turkey becoming a dictatorship) from the US and EU mainstream media before the 'coup' happened served nothing more as a way to justify an eventual coup in Turkey in the eyes of the West and some segment of Turkey. Had the 'coup' succeeded, no one would have bat an eye considering 'Erdogan was just a dictator anyway', sweeping aside that he and his party are popular and have been elected since 2002 constantly.
Gulen, who lives in the US, acquired a green card back in the 90's with the help, through letters of recommendation, of 2 former CIA members, Graham Fuller and another guy whose name i have forgotten, and a former US ambassador to Turkey. Getting help of 2 former CIA members is interesting and raises question marks. Furthermore, am not surprised that Graham Fuller was defensive of Gulen's organization after the failed 'coup' attempt.
Turkey is not the old Turkey anymore that simply nodded yes to every US/EU/NATO request and that only merely served as a NATO outpost and buffer state for the EU. Under Erdogan, despite his flaws, Turkey is making positive strives in many fields, but above all he is straightforward and cares about our country's interest, unlike many of the previous govts in the past. Had Erdogan chosen to be a puppet of the US, Turkey right now would not have faced these problems. I can bet good money on that.
Thanks to the failed 'coup' attempt by the Gulenists and the stance of the US and EU politicians and mainstream media, the Turkish society has become united, which was really polarized for a long time before the 'coup' attempt. Before this 'coup attempt, it was mostly the AKP and MHP supporters that were fervent opponents of the US, but you know you messed up if even pro-Western Turkish politicians and media are questioning and are alarmed by the US' role now. Anti-Americanism and anti-EU will without doubt reach new heights while Russia, China and Iran have won points in the eye of the public (and most probably our politicians too) for their supportive stance.
So, while there is no hard evidence (yet?), connecting developments in the past and now gives a good indication that the CIA most probably was involved in a 'coup' in Turkey once again.
Time to build new alliances.