Left-transitively homomorph-containing subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is left-transitively homomorph-containing if, for any group $$K$$ containing $$G$$ as a defining ingredient::homomorph-containing subgroup, $$H$$ is also a homomorph-containing subgroup of $$K$$.

Stronger properties

 * Weaker than::Fully invariant direct factor:

Weaker properties

 * Stronger than::Homomorph-containing subgroup
 * Stronger than::Fully invariant subgroup