Prehomomorph-contained subgroup of finite group

Definition
A subgroup of a group is termed a prehomomorph-contained subgroup of finite group if the whole group is a finite group and the subgroup is a prehomomorph-contained subgroup of it -- for any surjective homomorphism from another subgroup to it, it is contained in the other subgroup.

Weaker properties

 * Stronger than::Isomorph-free subgroup of finite group
 * Stronger than::Characteristic subgroup of finite group
 * Stronger than::Normal subgroup of finite group