# Center-fixing implies central factor-extensible

Suppose $G$ is a group and $\sigma$ is an automorphism of $G$ such that $\sigma$ is center-fixing: $\sigma(z) = z$ for all $z$ in the center of $G$. Then, $\sigma$ is a central factor-extensible automorphism of $G$. In other words, given any group $K$ containing $G$ as a central factor, $\sigma$ can be extended to an automorphism of $K$.
Note that this in particular shows that if every automorphism of $G$ is center-fixing (e.g., if $G$ is centerless, or has a center of order two), then $G$ is an AEP-subgroup inside any group in which it is a central factor.