This may not add much, but: If 0 -> Zr -> Zs -> G -> 0 is a free resolution of your group, then you can construct a fibration sequence K(Zs,n) -> K(G,n) -> K(Zr,n+1). You can then get something like the "factorizations" you're trying to interpret from the universal coefficient theorem that way, but it's from a somewhat less-standard universal coefficient theorem for cohomology
0 -> Hn(X;Z) ⊗ G -> Hn(X;G) -> Tor(Hn+1(X;Z),G) -> 0
(and I'm worried that maybe G has to be finitely generated). The cohomology classes on the tensor side come from those maps that lift from [X,K(G,n)] to [X,K(Zs,n)] up the fibration sequence, and the ones on the Tor side are what are left.