It seems likely that every tightly clique is trying to infiltrate every other such clique - it's endless battle between mafias, political parties, cults (Tulsi Gabard's connections to Krishna cult), intelligence agencies and so-forth, each trying to use the other.

But naturally, there significant limits on how much and how long each of infiltration be effective. A infiltrator from X sent to gain control of Y and gaining complete control there of will often identify with Y since leading it give them more power (Stalin was likely a agent of the Czarist secret police before the revolution but he probably wasn't taking orders from them in 1935 etc).

Now I want to play Steve Jackson's Illuminati...

https://www.sjgames.com/illuminati/