When is the automorphism group of a finite $p$-group nilpotent?

Question

Suppose $G$ is a finite $p$-group with odd $p$. Is it true, that $Aut(G)$ is nilpotent iff $G$ is cyclic?

When $G$ is cyclic, $Aut(G)$ is indeed abelian and thus nilpotent.

However, I do not know how to prove the statement that if $Aut(G)$ is nilpotent, then $G$ is cyclic. Nor do I possess any counterexamples.

Any help will be appreciated.

Answer 1

The order of the automorphism group of ${\tt SmallGroup}(31,729)$ is $3^9$ and hence nilpotent.

Answer 2

Perhaps it might be noted that it has been proved that in a suitable sense the automorphism group of a finite $p$-group is almost always a $p$-group.

As for concrete examples, see for instance this paper of mine and the references therein.