Congruent group extensions

Definition
Suppose $$A$$ and $$B$$ are (possibly isomorphic, possibly non-isomorphic groups). Consider two group extensions $$G_1, G_2$$ both "with normal subgroup $$A$$ and quotient group $$B$$." Explicitly, this means we are given two short exact sequences:

$$1 \to A \to G_1 \to B \to 1$$

and:

$$1 \to A \to G_2 \to B \to 1$$

We say that the group extensions are congruent if there is an isomorphism between the short exact sequences that restricts to the identity maps on $$A$$ and $$B$$ respectively. Explicitly, this means that there is an isomorphism $$\varphi: G_1 \to G_2$$ such that the following diagram commutes:

$$\begin{array}{lllll} 1 \to & A \to & G_1 \to & B \to & 1 \\ \downarrow & \downarrow^{\operatorname{id}_A} & \downarrow^{\varphi} & \downarrow^{\operatorname{id}_B} & \downarrow\\ 1 \to & A \to & G_2 \to & B \to & 1 \\ \end{array}$$

Related notions

 * Pseudo-congruent group extensions: This is a coarser equivalence relation, where we allow the maps on $$A$$ and $$B$$ to be automorphisms instead of requiring them to both be the identity map.