Prehomomorph-contained subgroup

Definition with symbols
Suppose $$H$$ is a subgroup of a group $$G$$. We say that $$H$$ is prehomomorph-contained in $$G$$ if for any surjective homomorphism of groups $$\varphi:K \to H$$ from a subgroup $$K$$ of $$G$$, we have $$H \le K$$.

Weaker properties

 * Stronger than::Intermediately strictly characteristic subgroup
 * Stronger than::Strictly characteristic subgroup
 * Stronger than::Isomorph-free subgroup
 * Stronger than::Isomorph-containing subgroup
 * Stronger than::Intermediately injective endomorphism-invariant subgroup
 * Stronger than::Injective endomorphism-invariant subgroup
 * Stronger than::Intermediately characteristic subgroup
 * Stronger than::Characteristic subgroup
 * Stronger than::Prehomomorph-dominated subgroup