CS-extensible automorphism

Symbol-free definition
An automorphism of a group is termed CS-extensible if it can be extended to an inner automorphism for every embedding of the group into the automorphism group of a characteristically simple group.

Definition with symbols
Let $$G$$ be a group and $$\sigma$$ an automorphism of $$G$$. We say that $$\sigma$$ is CS-extensible if for any embedding $$G \le Aut(N)$$ where $$N$$ is a characteristically simple group, there is an inner automorphism $$\phi$$ of $$Aut(N)$$ such that the restriction of $$\phi$$ to $$G$$ is $$\sigma$$.

Stronger properties

 * Extensible automorphism
 * Semidirectly extensible automorphism
 * Potentially characteristic-semidirectly extensible automorphism
 * CS-pushforwardable automorphism