Central factor-extensible automorphism

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This is a variation of extensible automorphism|Find other variations of extensible automorphism |

Definition

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 $\sigma$.

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.

Further information: Centerless implies every automorphism is central factor-extensible