Normal-isomorph-automorphic subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed a normal-isomorph-automorphic subgroup if $$H$$ is a defining ingredient::normal subgroup of $$G$$ and for any normal subgroup $$K$$ of $$G$$ isomorphic to $$H$$, $$H$$ and $$K$$ are automorphic subgroups in $$G$$, i.e., there is an automorphism $$\sigma$$ of $$G$$ such that $$\sigma(H) = K$$.