External central product

Definition
Suppose $$A$$ and $$B$$ are groups. Suppose we identify a central subgroup $$C$$ of $$A$$ with a central subgroup $$D$$ of $$B$$ via an isomorphism of groups $$\varphi:C \to D$$. The external central product of $$A$$ and $$B$$ with respect to $$\varphi$$ is the quotient of the external direct product $$A \times B$$ by the subgroup $$\{ g,\varphi(g)^{-1} \mid g \in C \}$$.

In particular, this is a group $$G$$ that has normal subgroups $$A_1$$ and $$B_1$$isomorphic to $$A$$ and $$B$$ respectively, such that $$A_1B_1 = G$$, $$A_1$$ and $$B_1$$ centralize each other, and $$A_1 \cap B_1$$ is like $$C \le A$$ when viewed as a subgroup of $$A_1$$ and like $$D \le B$$ when viewed as a subgroup of $$B_1$$. This is basically the definition of internal central product.