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

Statement
Suppose $$G$$ is a finite nilpotent group. Suppose the prime factors of $$G$$ are $$p_1,p_2,\dots,p_r$$ and the corresponding Sylow subgroups are respectively $$P_1,P_2,\dots,P_r$$. Then, the outer automorphism group $$\operatorname{Out}(G)$$ is isomorphic to the external direct product of the outer automorphism groups $$\operatorname{Out}(P_i)$$. Explicitly, in the corresponding internal direct product, the direct factor corresponding to $$\operatorname{Out}(P_i)$$ is the subgroup of $$\operatorname{Out}(G)$$ comprising those outer automorphism classes that fix all the $$P_j$$ for $$j \ne i$$.

Related facts

 * Automorphism group of finite nilpotent group is direct product of automorphism groups of Sylow subgroups