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 $H_{1}\leq G_{1}$ and $H_{2}\leq G_{2}$ 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 $H_{2}$.