Characteristic-semidirectly extensible automorphism

Definition with symbols
Let $$\sigma$$ be an automorphism of a group $$G$$. We say that $$\sigma$$ is characteristic-semidirectly extensible or a CSE-automorphism if the following holds:

Let $$\rho:G \to \operatorname{Aut}(N)$$ be a homomorphism such that $$N$$ is a characteristic subgroup of the associated semidirect product $$M = N \rtimes G$$. Then, there exists an automorphism $$\phi$$ of $$N$$ whose restriction to $$G$$ is $$\sigma$$.

Stronger properties

 * Extensible automorphism
 * Retraction-extensible automorphism
 * Semidirectly extensible automorphism
 * Potentially characteristic-semidirectly extensible automorphism

Weaker properties
For a finite group, linearly pushforwardable automorphism over a prime field where the prime does not divide the order of the group.