This may not add much, but: If $0 \to \mathbb{Z}^r \to \mathbb{Z}^s \to G \to 0$ is a free resolution of your group, then you can construct a fibration sequence $K(\mathbb{Z}^s, n) \to K(G, n) \to K(\mathbb{Z}^r, n+1)$. 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
$$0 \to H^n(X; \mathbb{Z}) \otimes G \to H^n(X; G) \to \operatorname{Tor}(H^{n+1}(X; \mathbb{Z}), G) \to 0$$
(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 $[X, K(\mathbb{Z}^s, n)]$ up the fibration sequence, and the ones on the $\operatorname{Tor}$ side are what are left.
