Unlocking the HQ is is straightforward for every other cult. Apprehension -> safehouse -> outpost, at each one capture the highest ranking member alive (and with so many of these missions spawning, one will capture them alive even if not really trying). However, Exalt doesn't seem to have the apprehension/safehouse/outpost chain.
I looked up the "Exalt Operations" research, it requires a "brainer". However, I never found any of them on any special missions featuring Exalt. What did I miss?