Even if it were a good idea, I don’t think this is the time to do it. Overlooked is the fact that the big Shiite leader in Baghdad has called not only for Shiites but for Sunnis and Kurds to fight against ISIL. Also, ISIL was perfectly happy to kill Sunnis in the (Sunni) area that they swept into first. There probably is a fair amount of unity among people opposed to ISIL.
This fight is only marginally connected to the conflicts between the Sunnis and the Shiites. It’s really an attack upon the very concept of a secular civil government.
I bet the “coalition” that Kerry wants, in all of his naivete (or ill-will) would actually be something where ISIL, which is an international terrorist group and not even Iraqi, would have a dominant voice. In other words, hand over Iraq to AQ while chirping about a “unified government.”
And how could the US or Iraq weed them out? Until ISIS took over a huge chunk of Iraq, we were still arming, funding and training them as "freedom fighters" against Assad (whom we should be working with by now). Their poster boy is that photo-op they had with McCain.
It's just good crisis management. Taking advantage of a situation is what we do, but in a real out of control crisis don't make unnecessary changes with unknown consequences.