Automorphism group of periodic nilpotent group is direct product of automorphism groups of Sylow subgroups

Statement
Suppose $$G$$ is a periodic nilpotent group. We know that this means that $$G$$ is a restricted external direct product of nilpotent p-groups. The automorphism group of $$G$$ is isomorphic to the (unrestricted) external direct product of the automorphism groups of each of the nilpotent $$p$$-groups.

Corollaries

 * Outer automorphism group of periodic nilpotent group is direct product of outer automorphism groups of Sylow subgroups