Definition:Pullback of Quotient Group Isomorphism

Definition
Let $G, H$ be groups.

Let $N \lhd G, K \lhd H$.

Let $G / N \cong H / K$ such that $\theta: G / N \to H / K$ is such an isomorphism.

The pullback $G \times^\theta H$ of $G$ and $H$ via $\theta$ is the subset of $G \times H$ of elements of the form $\left({g, h}\right)$ where $\theta \left({g N}\right) = h K$.

Also see

 * Pullback is Subgroup