Group of IIP 2-cocycles for trivial group action is naturally identified with group of homomorphisms from a particular abelian group

Statement
Suppose $$G$$ is a group. Then, there exists an abelian group $$K$$ such that, for any abelian group $$A$$, the group $$Z^2_{IIP}(G,A)$$ of IIP 2-cocycles $$\! f:G \times G \to A$$ for the trivial group actioncan be identified with the group of homomorphisms $$\operatorname{Hom}(K,A)$$ under pointwise addition.