Left-transitively isomorph-containing subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a left-transitively isomorph-containing subgroup if, whenever $$G$$ is an isomorph-containing subgroup of a group $$K$$, $$H$$ is also an isomorph-containing subgroup of $$K$$.

Weaker properties

 * Stronger than::Isomorph-containing subgroup
 * Stronger than::Characteristic subgroup
 * Stronger than::Normal subgroup