2-coboundary for a group action

Definition
Let $$G$$ be a group acting on an Abelian group $$A$$. A 2-coboundary for the action of $$G$$ on $$A$$, is a function $$f: G \times G \to A$$ such that there exists a function $$\phi:G \to A$$ such that:

$$f = (g,h) \mapsto g.\phi(h) - \phi(gh) + \phi(g)$$

Importance
Suppose we want to classify all groups $$E$$ which arise as the group extension with normal subgroup $$A$$ and quotient $$G$$. One approach to describing such a group $$E$$ is to define a collection $$S$$ of coset representatives for $$A$$ in $$E$$. This can be viewed as a map from $$G$$ to $$E$$. Call the coset representative for $$g$$ $$b_g$$. Define $$f(g,h)$$ as the element of $$A$$ given by $$b_{gh}^{-1}b_gb_h$$.

It turns out that if we pick two different collections of coset representatives and let $$f_1$$ and $$f_2$$ be the functions corresponding to them, then $$f_1 - f_2$$ is a 2-coboundary.

It's also true that each of $$f_1$$ and $$f_2$$ needs to be a 2-cocycle, and thus the collection of possible extensions of $$G$$ by $$A$$ is classified by the second cohomology group for the action of $$G$$ on $$A$$.