# Semidirectly 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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## Contents

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## Definition

### Definition with symbols

Let $\sigma$ be an automorphism of a group $G$. Then $\sigma$ is said to be semidirectly extensible if the following holds. For any homomorphism $\rho:G \to Aut(N)$ for a group $N$, consider the semidirect product $M = N \rtimes G$. Then, there exists an automorphism $\varphi$ of $M$ that leaves both $N$ and $G$ invariant, and whose restriction to $G$ is $\sigma$.