Berkovich's question on whether a group of prime power order has an outer automorphism of the same prime order

Statement
Suppose $$G$$ is a nontrivial group of prime power order that is not cyclic of prime order. Let $$p$$ be the prime. Does $$G$$ have an outer automorphism (i.e., an automorphism that is not an inner automorphism) that has order $$p$$?

It is known that the outer automorphism group contains an element of order $$p$$; in other words, there exists an outer automorphism whose $$p^{th}$$ power is in the inner automorphism group. Berkovich's question asks whether we can in fact find an outer automorphism for which the $$p^{th}$$ power is the identity element.

Related facts

 * Group of prime power order is either of prime order or has outer automorphism class of same prime order

Related open problems

 * Conjecture that the automorphism group of a non-cyclic group of prime power order is always bigger