From: Bettina Eick 
Date: Tue, 22 Apr 1997 17:41:41 +0200 (MET DST)

>Dear Forum,
>
>It is well known that the second cohomology group H^2(G,A)
>classifies extensions of G by A.
>When two elements of H^2(G,A) 
>corresponds to nonisomorphic groups?
>
>Alexei Davydov
 
I don't have a general solution for this question. There is the
following partial solution to the problem.
 
Suppose A is elementary abelian.
Let H_1 and H_2 be two extensions of G by A. Let A_i < H_i be
the subgroup corresponding to A in H_i for i = 1,2. Then we call
H_1 and H_2 stongly isomorphic, iff there exists an isomorphism 
H_1 -> H_2 which maps A_1 onto A_2. 
The stong isomorphism classes of extensions can be classified.
 
Let phi : G -> Aut(A) the operation homomorphism.
We call a tuple ( a, b ) of Aut(G) x Aut( M ) a compatible pair,
iff phi(g^a) = phi(g)^b for all g in G. The subgroup of all compatible
pairs is denoted by C.
 
There exists an action of C on H^2(G,A). Let psi in Z^2(G,A) and
(a,b) in C. Then we define psi^(a,b)(g,h) := psi(g^(a^-1), h^(a^-1))^b.
This defines an action of C on Z^2(G,A) which learves B^2(G,A)
invariant. Thus we obtain an action of C on H^2(G, A). (This action
is linear.) 
 
The strong isomorphism classes of extensions correspond one-to-one to the 
orbits of C on H^2(G, A) under this action.
 
Best wishes, Bettina Eick