Control as in expanding into other systems and being the dominate faction. How is it any different from Power Play?
That's the role of a minor faction within the BGS, so why not pick one, call its base home and support its expansion? FD have already said that minor factions can become powers, so if you put in the effort you could take it all the way. Who knows, if you petitioned FD with the support of enough players they might even create a minor faction with a group-nominated name that doesn't have a colour in it
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But that isn't what you mean, is it? You want a player group injected into the BGS with powers effectively to direct the rules governing the BGS in the systems they control. Problem is, that would destabilise the BGS and I doubt FD would ever cede you that much control over their creation.