Intermediately isomorph-conjugate subgroup

Symbol-free definition
A subgroup of a group is termed intermediately isomorph-conjugate if it is isomorph-conjugate in every intermediate subgroup.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed intermediately isomorph-conjugate if given any other subgroup $$L$$ of $$G$$ which is isomorphic to $$H$$, there is an element $$x \in $$ such that $$xHx^{-1} = L$$.

Stronger properties

 * Weaker than::Sylow subgroup in a finite group
 * Hall subgroup in a finite solvable group
 * Weaker than::Isomorph-free subgroup

Weaker properties

 * Stronger than::Intermediately isomorph-conjugate subgroup of normal subgroup
 * Stronger than::Pronormal subgroup
 * Stronger than::Weakly pronormal subgroup
 * Stronger than::Paranormal subgroup
 * Stronger than::Polynormal subgroup
 * Stronger than::Intermediately automorph-conjugate subgroup
 * Stronger than::Intermediately normal-to-characteristic subgroup
 * Stronger than::Intermediately subnormal-to-normal subgroup
 * Stronger than::Isomorph-conjugate subgroup

Metaproperties
If $$H$$ is intermediately isomorph-conjugate in $$G$$, then $$H$$ is also intermediately isomorph-conjugate in any intermediate subgroup. This follows from the definition.

If $$H, K \le G$$ are intermediately isomorph-conjugate subgroups, and $$K \le N_G(H)$$, then the join of subgroups $$HK$$ is also an intermediately isomorph-conjugate subgroup.