Central factor-extensible automorphism

Symbol-free definition
An automorphism of a group is termed central factor-extensible if, for every embedding of the group as a central factor of a group, the automorphism can be extended to an automorphism of the bigger group.

Definition with symbols
An automorphism $$\sigma$$ of a group $$G$$ is termed central factor-extensible if, for every embedding of $$G$$ as a central factor in a group $$H$$, there exists an automorphism $$\varphi$$ of $$H$$ whose restriction to $$G$$ is $$\sigma$$.

Stronger properties

 * Weaker than::Extensible automorphism
 * Weaker than::Normal-extensible automorphism
 * Weaker than::Center-fixing automorphism:

Facts
In a centerless group, every automorphism is central factor-extensible. This is because any central factor that is centerless as a group must be a direct factor.