Group of cyclicity-preserving 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 of cyclicity-preserving 2-cocycles for the trivial group action $$\! f:G \times G \to A$$ can be identified with the group of homomorphisms $$\operatorname{Hom}(K,A)$$ under pointwise addition.

Extreme examples
If $$G$$ is a cyclic group or more generally a locally cyclic group, the corresponding group $$K$$ is a trivial group.

Examples of 2-groups
When $$G$$ is a 2-group, $$K$$ is also a 2-group.

Examples of other p-groups
We consider the case $$\! p = 3$$.