Image-closed fully invariant subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed image-closed fully invariant in $$G$$ if, for any surjective homomorphism $$\varphi:G \to K$$, $$\varphi(H)$$ is a fully invariant subgroup of $$K$$.

Stronger properties

 * Weaker than::Verbal subgroup
 * Weaker than::Normal Sylow subgroup
 * Weaker than::Normal Hall subgroup
 * Weaker than::Order-containing subgroup

Weaker properties

 * Stronger than::Fully invariant subgroup
 * Stronger than::Image-closed characteristic subgroup
 * Stronger than::Characteristic subgroup