Right-transitively homomorph-containing subgroup

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

Stronger properties

 * Weaker than::Subhomomorph-containing subgroup:

Weaker properties

 * Stronger than::Homomorph-containing subgroup