Potentially characteristic-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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Definition

Definition with symbols

Let $\sigma$ be an automorphism of a group $G$. Then $\sigma$ is said to be potentially characteristic-semidirectly extensible if the following holds:

Let $\rho :G\to Aut(N)$ be a homomorphism such that $N$ is a potentially characteristic subgroup of the semidirect product $M$ of $N$ with $G$. Then, there exists an automorphism $\phi$ of $M$ that leaves both $N$ and $G$ invariant, and whose restriction to $G$ is $\sigma$.

Relation with other properties

Stronger properties

• Extensible automorphism: For full proof, refer: extensible implies potentially characteristic-semidirectly extensible
• Semidirectly extensible automorphism

Weaker properties

• Characteristic-semidirectly extensible automorphism
• CS-pushforwardable automorphism: The implication follows from the automorphism group action lemma
• CS-extensible automorphism