Pullback of Quotient Group Isomorphism is Subgroup

Theorem
Let $G, H$ be groups.

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

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

Let $G \times^\theta H$ be the pullback of $G$ and $H$ via $\theta$.

Then $G \times^\theta H$ is a subgroup of $G \times H$.