Nilpotent p-group and every automorphism is inner implies trivial or cyclic or order two

Statement
Suppose $$G$$ is a (possibly finite, possibly infinite) group that is a nilpotent group, a p-group, and a group in which every automorphism is inner. Then, $$G$$ must be either the trivial group or cyclic group:Z2.

Related facts

 * Finite nilpotent and every automorphism is inner implies trivial or cyclic of order two
 * Finitely generated nilpotent and every automorphism is inner implies trivial or cyclic of order two
 * There exist infinite nilpotent groups in which every automorphism is inner