# Characteristic-extensible automorphism

This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
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This is a variation of extensible automorphism|Find other variations of extensible automorphism |
This term is related to: Extensible automorphisms problem
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## Definition

### Symbol-free definition

An automorphism of a group is said to be characteristic-extensible if, for every embedding of the group as a characteristic subgroup of a bigger group, the automorphism can be extended to the bigger group.

### Definition with symbols

An automorphism $\sigma$ of a group $G$ is said to be characteristic-extensible if for any embedding of $G$ as a characteristic subgroup of a group $H$, there exists an automorphism $\phi$ of $H$ such that the restriction of $\phi$ to $G$ is $\sigma$.

## Formalisms

### In terms of the qualified extensibility operator

This property is obtained by applying the qualified extensibility operator to the property: characteristicity
View other properties obtained by applying the qualified extensibility operator

The property of being a characteristic-extensible automorphism is obtained by applying the qualified extensibility operator, where the automorphism property is the tautology, and the subgroup property is that of being characteristic.