Cocycles and nontrivial central extensions

Question

Let $G$ be a group and $A$ an Abelian group. In group cohomology, it's a well-known fact that given two cocycles $\omega_1, \omega_2:G\times G\to A$, the central group extensions $G\times_{\omega_i} A$ defined by them are equivalent if and only if the cocycles are equivalent, i.e. they differ by a coboundary.

On the other hand, two central extensions, $A\to E_1\to G$ and $A\to E_2\to G$ can be inequivalent in different manners:

1. $E_1$ and $E_2$ are not isomorphic
2. $E_1$ and $E_2$ are isomorphic, say both are $E$, but the embedding $A\to E$ is different in the two extensions.
3. $E_1$ and $E_2$ are isomorphic, say both are $E$, but the projection $E\to G$ is different in the two extensions.
4. $E_1$ and $E_2$ are isomorphic, say both are $E$, but both the embedding $A\to E$, both the projection $E\to G$ is different in the two extensions.

Given $G$, $A$, and a cocycle $\omega:G\times G\to A$ that is not a coboundary, is there a general way to determine whether the central extension $G\times_{\omega} A$ defined by $\omega$, in what manner differs from the direct product $G\times A$?

Edit

As Derek Holt pointed out, my question has sense only if $G\times A$ can be isomorphic with a nonsplit central extension of $G$ by $A$ (sorry for writing $G$ by $A$ instead of $A$ by $G$, but I prefer this terminology, as nLab convinced me of doing this). So, my first question is: Can $G\times A$ be isomorphic with a nonsplit central extension of $G$ by $A$?

Answer 1

This answer is just a summary of the discussion in the comments to this question on MathOverflow.

It is proved here (J. Ayoub, The direct extension theorem. J. Group Theory 9 (2006), 307-316) that a direct product $G \times A$ of finite groups cannot be isomorphic to a nonsplit extension of $G$ by $A$ (with either interpretation).

But there are easy infinite examples. Let $H = C_2 \times C_4$, and let $G$ be a direct product of countably infinitely many copies of $H$. Then $G \cong G \times G$, but by splitting one or more of the $C_4$ factors in $G$, we see that $G$ can be expressed in many different ways as a nonsplit extension of $G$ by $G$.