The Gulenist movement was very involved in education. They had maybe hundreds of schools in every category of Turkish education system, and they have some Turkish language schools abroad too, and also provide loads of scholarship. The general opinion is that through these schools they were raising loyal members of the movement who'd go into all sorts of public and private institutions as high-rank employees. This opinion exists at least since 90's, and is vastly popular. It's known that members pay a fixed amount (AFAIK a percentage of their monthly pay) monthly to the organisation. Most members are conservative celebs, conservative enterpreneurs and white collar types.

After their break-up with AKP, when the corruption scandals showed up, the movement quickly became the scapegoat of the ruling party, accused of having formed a "parallel statal system". IDK if Gulen himself organised this coup (how can I know), but I can say that, reasoning on the Turkish public's view of the movement, including third party stories I've heard over the years (tho I've never met a member), they certainly have the network, hierarchy and the power to do it. It may also be someone in their hierarchy below Gulen that organised it. Time will show us the reality, I think.