CS-pushforwardable automorphism

Definition with symbols
Let $$G$$ be a group and $$\sigma$$ an automorphism of $$G$$. $$\sigma$$ is said to be CS-pushforwardable if for any homomophism $$\rho:G \to Aut(N)$$ where $$N$$ is a characteristically simple group, there exists an inner automorphism $$\phi$$ of Aut(N) such that $$\phi \circ \rho = \rho \circ \sigma$$.

Stronger properties

 * Extensible automorphism
 * Semidirecty extensible automorphism
 * Potentially characteristic-semidirectly extensible automorphism: The implication is a consequence of the automorphism group action lemma

Weaker properties

 * CS-extensible automorphism