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 $$G \times H$$.