Pseudo-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. Explicitly, this means that there are automorphisms $$\alpha \in \operatorname{Aut}(A)$$, $$\beta \in \operatorname{Aut}(B)$$, and an isomorphism $$\varphi: G_1 \to G_2$$ such that the following diagram commutes:

$$\begin{array}{lllll} 1 \to & A \to & G_1 \to & B \to & 1 \\ \downarrow & \downarrow^{\alpha} & \downarrow^{\varphi} & \downarrow^{\beta} & \downarrow\\ 1 \to & A \to & G_2 \to & B \to & 1 \\ \end{array}$$

Related notions

 * Congruent group extensions: This is a finer equivalence relation imposed on group extensions, where we require the automorphisms of $$A$$ and $$B$$ to both be identity maps.