Original Question:
Let $G$ be a group and $N \trianglelefteq G$. $N$ is abelian. If $\gcd(|G/N|,|\text{Aut}(G)|)=1$, then prove that $N \subseteq Z(G)$.
All I could find is that Inner Automorphisms of $N$ is trivial and the automorphism group is of the form $|N|k, k\in \mathbb{N}.$ I am feeling kind of lost and just wanted a push in the right direction. Thank you in advance.