Automorphism groups of 2-groups

A well-known conjecture on p-groups states that every non-abelian p-group G has the property that |G| divides |Aut(G)|. We exhibit periodic patterns in the automorphism group orders of the 2-groups of fixed coclass and we use this to show that for every positive integer r there are at most finitely many counterexamples to the conjecture among the 2-groups of coclass r.