Automorphism-faithful subgroup

Symbol-free definition
A subgroup of a group is termed automorphism-faithful if any nontrivial automorphism of the whole group that restricts to an automorphism of the subgroup, in fact restricts to a nontriival automorphism of the subgroup.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed automorphism-faithful in $$G$$ if for any nontrivial automorphism $$\sigma$$ of $$G$$ such that $$\sigma(H) = H$$, the restriction of $$\sigma$$ to $$H$$ is a nontrivial automorphism of $$H$$.

Stronger properties

 * Weaker than::Centralizer-free normal subgroup:
 * Weaker than::Centralizer-free characteristic subgroup
 * Weaker than::Automorphism-faithful normal subgroup
 * Weaker than::Automorphism-faithful characteristic subgroup

Weaker properties

 * Stronger than::Coprime automorphism-faithful subgroup