Sure. The Big Ten had Rutgers on their radar since Pernetti was a freshman in college. But that doesn't mean that Pernetti actually did anything to move the Big Ten to extend the invitation.
If Pernetti was actively involved in discussions with the Big Ten, he certainly would have known in January that an invitation was pending. Pitt, Syracuse, and WVU had already decided to leave the Big East. Pernetti, if he were active, certainly would have had discussions with Delany about timing of a Big Ten invite versus remaining in a crumbling Big East. Maybe Pernetti didn't know that the invite would come in November vs the following year, but if he was in active discussions with the Big Ten, he would have known the invitation was going to happen soon and certainly within the tenure of whatever coach was hired to replace Schiano.
To claim that Pernetti was hamstrung by the crumbling Big East is to accept the narrative that he was just a passive observer in the Big Ten invitation, and that his role was essentially no more than "answering the phone".
If you believe that Pernetti had an active role in getting Rutgers into the Big Ten, then you can't really accept the claim that the crumbling Big East was a significant factor in his coach hiring decision.