It is a basic result of group cohomology that the extensions with a given abelian normal subgroup A and a given quotient G acting on it via an action $\varphi$ are given by the second cohomology group $H^2_\varphi(G,A)$. In particular, when the action is trivial (so the extension is a central extension), this is the second cohomology group $H^2(G,A)$ for the trivial action. In the special case where G is also abelian, we classify all the class two groups with A inside the center and G as the quotient group.

I am interested in the following: given a sequence of abelian groups $A_1, A_2, \dots, A_n$, what would classify (up to the usual notion of equivalence via commutative diagrams) the following: a group E with an ascending chain of subgroups:

$$1 = K_0 \le K_1 \le K_2 \le \dots \le K_n = E$$

such that the $K_i$s form a central series (i.e., $[E,K_i] \subseteq K_{i-1}$ for all i) and $K_i/K_{i-1} \cong A_i$?

The case $n = 2$ reduces to the second cohomology group as detailed in the first paragraph, so I am hoping that some suitable generalization involving cohomology would help describe these extensions.

Note: As is the case with the second cohomology group, I expect the object to classify, not isomorphism classes of possibilities of the big group, but a notion of equivalence class under a congruence notion that generalizes the notion of congruence of extensions. Then, using the actions of various automorphism groups, we can use orbits under the action to classify extensions under more generous notion of equivalence.

Note 2: The crude approach that I am aware of involves building the extension step by step, giving something like a group of groups of groups of groups of ... For intsance, in the case $n = 3$:

$$1 = K_0 \le K_1 \le K_2 \le K_3 = G$$

with quotients $A_i \cong K_i/K_{i-1}$, I can first consider $H^2(A_3,A_2)$ as the set of possibilities for $K_3/K_1$ (up to congruence). For each of these possibilities P, there is a group $H^2(P,A_1)$ and the total set of possibilities seems to be:

$$\bigsqcup_{P \in H^2(A_3,A_2)} H^2(P,A_1)$$

Here the $\in$ notation is being abused somewhat by identifying an element of a cohomology group with the corresponding extension's middle group.

What I really want is some algebraic way of thinking of this unwieldy disjoint union as a single object, or some information or ideas about its properties or behavior.