So I have been recently investigating how Abelian group extensions work. The question we ask ourselves here is: Suppose $H$ and $N$ are groups, of which $N$ is abelian. Suppose we have an $H$-action given on $N$. How many exact sequences of the following form...
$0\longrightarrow N\longrightarrow G\longrightarrow H\longrightarrow 0$
... exist, up to iso, where the conjugationaction of $H$ (it is an $H$ action as $N$ is Abelian) on $N$ corresponds to the given action?
I have found out that the set of classes of such exact sequences, $Ext(H,N)$ has an Abelian group structure, which is the same as the Bear sum in the Abelian case. We actually obtain a functor $Ext(H,-): \textbf{Ab}_H\to \textbf{Ab}$ this way.
My questions are:
1) Is there some good literature about this?
2) Is it true that this is part of a long exact sequence as in the abelian case?
3) I found a canonical morphism $Ext(H_{ab}, N^{\phi})\to Ext(H, N)$ (where $\phi$ is the $H$-action). Under what conditions is this an isomorphism?