Intermediately automorph-conjugate subgroup

Symbol-free definition
A subgroup of a group is said to be intermediately automorph-conjugate if it is an automorph-conjugate subgroup in every intermediate subgroup (viz, every subgroup of the whole group containing it).

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is said to be intermediately automorph-conjugate if for any subgroup $$K$$ of $$G$$ such that $$H \le K$$, $$H$$ is an automorph-conjugate subgroup of $$K$$. In other words, for any automorphism $$\sigma$$ of $$K$$, there exists $$g \in K$$ such that $$\sigma(H) = gHg^{-1}$$.

Stronger properties

 * Weaker than::Isomorph-free subgroup
 * Weaker than::Intermediately isomorph-conjugate subgroup
 * Weaker than::Intermediately characteristic subgroup
 * Weaker than::Sylow subgroup

Weaker properties

 * Stronger than::Intermediately automorph-conjugate subgroup of normal subgroup
 * Stronger than::Weakly pronormal subgroup
 * Stronger than::Polynormal subgroup
 * Stronger than::Intermediately normal-to-characteristic subgroup
 * Stronger than::Intermediately subnormal-to-normal subgroup
 * Stronger than::Automorph-conjugate subgroup