Coprime automorphism-invariant subgroup

Definition
Suppose $$G$$ is a finite group and $$H$$ is a subgroup of $$G$$. We say that $$H$$ is a coprime automorphism-invariant subgroup of $$G$$ if, for any automorphism $$\sigma$$ of $$G$$ whose order is relatively prime to the order of $$G$$, we have $$\sigma(H) = H$$.

Stronger properties

 * Weaker than::Characteristic subgroup (in a finite group)
 * Weaker than::Coprime automorphism-invariant normal subgroup
 * Weaker than::Hall-relatively weakly closed subgroup