2-cocycle for trivial group action

Definition
Suppose $$G$$ is a group and $$A$$ is an abelian group. A 2-cocycle for trivial group action for $$G$$ on $$A$$ is a 2-cocycle for the trivial group action of $$G$$ on $$A$$.

Explicitly, it is a function $$f:G \times G \to A$$ satisfying the following condition:

$$\! f(g,hk) + f(h,k) = f(gh,k) + f(g,h) \ \forall \ g,h,k, \in G$$

The set of 2-cocycles for trivial group action form a group, denoted $$Z^2(G,A)$$. Note that the same notation is used for the group of 2-cocycles for a nontrivial group action as well, though in the latter case, a subscript for the action may be used or the specific action is made clear from the context.

Extensions involving a central subgroup
Let $$E$$ be a group with a central subgroup isomorphic to (and explicitly identified with) $$A$$, and a quotient isomorphic to (and explicitly identified with) $$G$$, such that the induced action of the quotient on the subgroup (in the sense of action by conjugation, see quotient group acts on abelian normal subgroup). Let $$S$$ be a system of coset representatives for $$G$$ in $$E$$ with $$s: G \to S$$ being the representation map. Then, define $$f: G \times G \to A$$ such that

$$\! s(gh) = f(g,h)s(g)s(h)$$

In other words, $$f$$ measures the extent to which the collection of coset representatives fails to be closed under multiplication.

Such an $$f$$ is a 2-cocycle for trivial group action of $$G$$ on $$A$$.

Note that for a particular choice of $$E$$, all the 2-cocycles obtained by different choices of $$S$$ will form a single coset of the coboundary group, that is, any two such cocycles will differ by a coboundary. Thus, in particular, we can intrinsically associate, to every extension $$E$$ with abelian normal subgroup $$A$$ and quotient $$G$$, an element of the second cohomology group.