Quotient-isomorph-automorphic subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a quotient-isomorph-automorphic subgroup if $$H$$ is a defining ingredient::normal subgroup of $$G$$ and, for any normal subgroup $$K$$ of $$G$$ such that $$G/H \cong G/K$$, there exists an automorphism $$\sigma$$ of $$G$$ such that $$\sigma(H) = K$$.

Stronger properties

 * Weaker than::Quotient-isomorph-free subgroup

Weaker properties

 * Stronger than::Normal subgroup