Action-isomorph-free subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed action-isomorph-free in $$G$$ if $$H$$ is a normal subgroup of $$G$$, and the following condition holds.

Suppose $$\alpha:G \to \operatorname{Aut}(H)$$ be the homomorphism induced by the action of $$G$$ on $$H$$ by conjugation. Suppose $$K$$ is a normal subgroup of $$G$$ with $$\beta:G \to \operatorname{Aut}(K)$$ the homomorphism induced by the action of $$G$$ on $$K$$ by conjugation. Suppose, further, that $$\sigma:H \to K$$ is an isomorphism with the property that: