I'm looking for an example of a finite abelian group <em>A</em> and a finite group <em>G</em> acting trivially on <em>A</em> such that there are two extensions $E_1$ and $E_2$ with base <em>A</em> and quotient <em>G</em> (i.e., they are both central extensions, and hence both give corresponding elements of $H^2(G,A)$) and:

 1. $E_1$ and $E_2$ are isomorphic as abstract groups.
 2. Under the natural action of $\operatorname{Aut}(G) \times \operatorname{Aut}(A)$ on $H^2(G,A)$ (by pre- and post-composition with 2-cocycles that then descends to action on cohomology classes), the cohomology classes corresponding to $E_1$ and $E_2$ are <em>not</em> in the same orbit.

Basically condition (2) states that $E_1$ and $E_2$ are not only not congruent extensions, they are not even congruent up to a relabeling of the subgroup <em>A</em> and the quotient <em>G</em>. Another way of putting this is that there is no isomorphism between $E_1$ and $E_2$ that sends the <em>A</em> inside $E_1$ to the <em>A</em> inside $E_2$.

The analogous statement with a nontrivial action of <em>G</em> on <em>A</em> is also of interest to me. In this latter case, though, the entire group $\operatorname{Aut}(G) \times \operatorname{Aut}(A)$ does not act.

I think that examples exist (because of my experience with finding examples for similar specifications) but there may well be a proof to the contrary.