Completely distinguished subgroup

Symbol-free definition
A subgroup of a group is termed completely distinguished if for every surjective endomorphism from the group to itself, the subgroup equals its complete pre-image.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed completely distinguished in $$G$$ if it satisfies the following equivalent conditions:


 * For any surjective endomorphism $$f:G \to G$$, $$f^{-1}(H) = H$$
 * $$H$$ is a distinguished subgroup (also termed a strictly characteristic subgroup) and $$G/H$$ is a Hopfian group