Since Mickey Mouse Corleoned out Prince Xizor, who runs Black Sun in Imperial Era according to Disney canon? (after Clone-War-Era Ziton Moj).
During the Clone Wars Era, the Black Sun was run by a group of Falleen noblemen from a fortress on Mustafar.
Alter during the Clone Wars, the Black Sun was assimilated (by the order of Darth Maul) into the Shadow Collective.
Since there is no mention of any leadership change at the Clone Wars end and during the Empire Era, Ziton Moj should still be the leader. He was not reported to be killed in the conflict with the Pyke Syndicate (specifically, Krim's family rescue mission on Mustafar) so it is logical to assume he remained in command.
Although there is no good answer, one can assume that the Empire definitely plays a large part in it due to their partnership with the group. I suggest watching Star Wars Rebels or reading Tarkin.