Left-transitively permutable subgroup

Symbol-free definition
A subgroup of a group is termed left-transitively permutable if whenever the group is embedded as a permutable subgroup of some bigger group, the subgroup is also permutable inside the bigger group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed left-transitively permutable if for any embedding of $$G$$ in some group $$K$$, such that $$G$$ is permutable inside $$K$$, $$H$$ is also permutable inside $$K$$.

Weaker properties

 * Stronger than::Characteristic subgroup:
 * Stronger than::Normal subgroup
 * Stronger than::Permutable subgroup