Carol was referring to Starfleet when she asked: "why are you taking Genesis from us?"
Why would she hunt Admiral Kirk down, on assignment aboard the Enterprise, instead of someone from Starfleet actually involved in the project?
Kirk had to be involved at some level or it makes no sense for Carol Marcus to go out of her way to him for information. Then you add the fact that he seemed pretty knowledgeable about the project and had security clearance to materials related to it.
Because Khan used Kirk's name to threated Genesis to lure Kirk into the situation. Kirk only had basic info on Genesis. Also why would he be involved when he was just in charge of or part of Starfleet Academy?
The fact that he had security clearance could be due to the fact that he was an admiral.