Center-fixing implies central factor-extensible

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