Automorphism group of a non_abelian p_group

Question

Let G be a non abelian p_group. When is set of all automorphisms group of G a p_group?

Answer 1

Depending on the power of $p$ there is a classification. For example, amongst all $p$-groups of order $p^5$, with $p>2$ prime, there is exactly one group whose automorphism group is again a $p$-group -- see here. It has order $p^6$. However, although it seems to be difficult to find finite $p$-groups for $p>2$ whose automorphism group is a $p$-group, there is an asymptotic result: The automorphism group of a finite p-group is almost always a p-group.