Coprime automorphism-faithful subgroup

Definition
A subgroup $$H$$ of a finite group $$G$$ is termed coprime automorphism-faithful if, given any non-identity automorphism $$\sigma \in \operatorname{Aut}(G)$$ such that $$\sigma(H) = H$$, and such that the order of $$\sigma$$ is relatively prime to the order of $$G$$, $$\sigma$$ acts nontrivially on $$H$$. Equivalently, any defining ingredient::coprime automorphism group of $$G$$ that acts trivially on $$G$$ must be trivial.

Relation with other properties

 * Weaker than::Coprime automorphism-faithful characteristic subgroup