Periodic nilpotent and every automorphism is inner implies trivial or cyclic of order two

Statement
Suppose $$G$$ is a periodic nilpotent group that is also a group in which every automorphism is inner. Then, $$G$$ must be either a trivial group or cyclic group:Z2.

Facts used

 * 1) uses::Nilpotent p-group and every automorphism is inner implies trivial or cyclic or order two
 * 2) uses::Outer automorphism group of periodic nilpotent group is direct product of outer automorphism groups of Sylow subgroups

Proof
The proof follows directly from Facts (1) and (2).