Center-fixing implies central factor-extensible

This article gives the statement and possibly, proof, of an implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., center-fixing automorphism) must also satisfy the second automorphism property (i.e., central factor-extensible automorphism)
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Statement

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.

Related facts

• Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible