# Semidirectly extensible automorphism

From Groupprops

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 term is related to: Extensible automorphisms problem

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## Definition

### Definition with symbols

Let be an automorphism of a group . Then is said to be **semidirectly extensible** if the following holds. For any homomorphism for a group , consider the semidirect product . Then, there exists an automorphism of that leaves both and invariant, and whose restriction to is .