Transfer-closed fully invariant subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed transfer-closed fully invariant if, for any subgroup $$K$$ of $$G$$, $$H \cap K$$ is a defining ingredient::fully invariant subgroup of $$K$$.

Stronger properties

 * Weaker than::Subhomomorph-containing subgroup
 * Weaker than::Variety-containing subgroup

Weaker properties

 * Stronger than::Intermediately fully invariant subgroup
 * Stronger than::Fully invariant subgroup
 * Stronger than::Transfer-closed characteristic subgroup
 * Stronger than::Intermediately characteristic subgroup
 * Stronger than::Characteristic subgroup
 * Stronger than::Normal subgroup