Injective endomorphism-quotient-balanced subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed injective endomorphism-quotient-balanced if $$H$$ is a normal subgroup of $$G$$ and every injective endomorphism of $$G$$ sends $$H$$ to itself and induces an injective endomorphism on the quotient group $$G/H$$.

Stronger properties

 * Weaker than::Finite injective endomorphism-invariant subgroup
 * Weaker than::Finite fully invariant subgroup
 * Weaker than::Characteristic subgroup of finite group
 * Weaker than::Injective endomorphism-invariant subgroup with injective endomorphism-invariant complement

Weaker properties

 * Stronger than::Injective endomorphism-invariant subgroup
 * Stronger than::Characteristic subgroup
 * Stronger than::Normal subgroup