Surjective endomorphism-balanced subgroup

Symbol-free definition
A subgroup of a group is termed surjective endomorphism-balanced if every surjective endomorphism of the whole group restricts to a surjective endomorphism of the subgroup.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed surjective endomorphism-balanced if for any surjective endomorphism $$\sigma$$ of $$G$$, $$\sigma(H) = H$$.

Formalisms
In the function restriction formalism, the property of being highly strictly characteristic is expressed as:

left side of function restriction expression::Surjective endomorphism $$\to$$ left side of function restriction expression::surjective endomorphism

Thus, it is a.

Stronger properties

 * Weaker than::Completely strictly characteristic subgroup

Weaker properties

 * Stronger than::Strictly characteristic subgroup (also called distinguished subgroup)
 * Stronger than::Right-transitively strictly characteristic subgroup
 * Stronger than::Characteristic subgroup

Metaproperties
Transitivity follows directly from the fact that the property is a balanced subgroup property.