Nilpotent group in which every automorphism is inner

Definition
A nilpotent group in which every automorphism is inner is a group that is both a nilpotent group and a group in which every automorphism is inner.

Any such group must be either the trivial group, or cyclic group:Z2, or an infinite (and infinitely generated) nilpotent group. See finitely generated nilpotent and every automorphism is inner implies trivial or cyclic of order two and there exist infinite nilpotent groups in which every automorphism is inner.