They mainly manage our satellites. Including communications, intelligence gathering, GPS, missile detection systems and monitor/protect them from attack/EM interference. That includes all of our civilian satellites.
They also make sure we (the whole planet) don't end up like the dinosaurs. Which is nice.
I'm guessing it would depend on how early they're detected. I'm not involved in planning or anything, just putting what I know about them out there. But yes, they would be who NASA calls when they find something.
I know they were involved in the DART mission. With the satellites used being launched and operated out of Vandenberg, a Space Force base.