Who's This? A shadowy organization.
How you could have heard of them: I dunno. Psychic powers?
Example story: The Daring New Adventures of Supergirl #7 (May 1983) "This Meeting Will Come to Disorder!" by Paul Kupperberg, Carmine Infantino and Bob Oksner
Meanwhile, Supergirl makes short work of the green-clad guards, but has a brief encounter with the Council's super-powered agents that prevents her from getting to the Council chamber in time to stop Project Skyeye from being launched off-site.
All Supergirl finds is Lester Adams' dead body, and she only recognizes him because someone called Johnny Ostrander (*raising eyebrow*) gave her a description. Shadowy, shadowy!
But of course, organizations like the Council are a dime a dozen in comics (just the fact that THIS Council is the third by THAT NAME in the DCU ALONE says something). I think they were also in Captain America: The Winter Soldier. So it kind of HAS to be about their cronies, or perhaps some interesting gimmick like the underwater base. So it must really hurt when a superhero or the authorities discover your secret HQ and you have to abandon it. How much did that place COST? Supergirl might as well have let the waters rush all the way in...
Who's Next? A tipsy aristocrat.