IIP 2-cocycle for trivial group action

Definition
Suppose $$G$$ is a group, $$A$$ is an abelian group, and $$f: G \times G \to A$$ is a function. We say that $$f$$ is an IIP 2-cocycle for trivial group action if it satisfies the following three conditions:

The group of IIP 2-cocycles is denoted $$Z^2_{IIP}(G,A)$$.