Automorphism group of finite nilpotent group is direct product of 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 automorphism group $$\operatorname{Aut}(G)$$ is isomorphic to the external direct product of the automorphism groups $$\operatorname{Aut}(P_i)$$. Explicitly, in the corresponding internal direct product, the direct factor corresponding to $$\operatorname{Aut}(P_i)$$ is the subgroup of $$\operatorname{Aut}(G)$$ comprising those automorphism that fix all the $$P_j$$ for $$j \ne i$$.

Corollaries

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