Equivalence of subgroups

Definition
Suppose $$G_1$$ and $$G_2$$ are groups, with $$H_1$$ a subgroup of $$G_1$$ and $$H_2$$ a subgroup of $$G_2$$. An equivalence of subgroups between the group-subgroup pairs $$H_1 \le G_1$$ and $$H_2 \le G_2$$ is an isomorphism of groups $$\sigma:G_1 \to G_2$$ such that the restriction of $$\sigma$$ to $$H_1$$ defines an isomorphism from $$H_1$$ to $$H_2$$.