Potentially characteristic-semidirectly extensible automorphism

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 $$\sigma$$.

Stronger properties

 * Extensible automorphism:
 * Semidirectly extensible automorphism

Weaker properties

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