Erdogan is aware that the British are behind the coup attempt
Erdogan is no doubt aware that the British have the power in the army, even though diminished, and that military loyal to British are behind the coup attempt. Nevertheless he accuses Gulen because talking about the pro-British men in the Army upholds them, and eliminating them without mentioning them trivializes their matter. Whereas Gulen is not just as significant as them ... Erdogan wants to eliminate the pro-British men without noise, that is in discreet, so that they do not stand out and there is no bloc around them ...
And in return wants to weaken his rival by noise because the Gulen Movement is not of the same power as the British men.
This is probably the opinion on what happened ... and all what happened was not a tightly planned coup attempt nor was it well thought out with poise, but is more of an outraged risk without provision and poise. The important thing is not to stop at what happened, but what is expected after that.