Cyclicity-preserving 2-cocycle for trivial group action

Definition
Suppose $$G$$ is a group and $$A$$ is an abelian group. A function $$f:G \times G \to A$$ is termed a cyclicity-preserving 2-cocycle for trivial group action if it satisfies the following conditions:

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

Existence of a source group
For any group $$G$$, there exists an abelian group $$K$$ such that for any abelian group $$A$$, the group of cyclicity-preserving 2-cocycles $$\! f:G \times G \to A$$ can be identified with the group $$\operatorname{Hom}(K,A)$$.

Extreme examples

 * If $$G$$ is a cyclic group or a locally cyclic group, the group of cyclicity-preserving 2-cocycles is a trivial group.
 * If $$A$$ is a trivial group, the group of cyclicity-preserving 2-cocycles is a trivial group.