Semidirectly extensible automorphism

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 $$\sigma$$.

Weaker properties

 * Retraction-extensible automorphism
 * Retraction-pullbackable automorphism
 * Characteristic-semidirectly extensible automorphism
 * Potentially characteristic-semidirectly extensible automorphism