I'm the new guy here, so take everything I say with a grain of caution that it might aswell be total baloney.
As I understood it, the name is decriptive more than anything, because nobody knows who they are.
When I picture them in my head I always think about Cyperpunk corporations. As in 6 filthy rich buisnessmen/women who wanna take over the 'underground' to protect their own buisnesses and to tap into those profits themselves.
A second possibility would be a rival Mafia organisation, like let's say the Yakuza? They don't want to risk an all out war because the loses would be too great, but instead look for other ways to destabilize the current power structure of the Deluca universe. Like kidnapping the children and trying to "reprogramm" them or pumping money into smaller families to motivate them to go for a bigger piece of the pie, thereby causing unrest and instability.
I might be wrong here, but all problems seem to originate from "smaller" families, like Straffan becoming too ambitious, Mina saying "I did what I had to do to protect our family", the Kaskars having a mole, a powerstruggle within the Bernero Family and Bonanzo Family. Now all of that might be buisness as usual, but if someone were to try to cause havoc and instability, well...that's how I would do it. Empower the weak and let them challenge the strong, then pick up the pieces.