This may not add much, but: If 0 -> Z<sup>r</sup> -> Z<sup>s</sup> -> G -> 0 is a free resolution of your group, then you can construct a fibration sequence K(Z<sup>s</sup>,n) -> K(G,n) -> K(Z<sup>r</sup>,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
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0 -> H<sup>n</sup>(X;Z) &otimes; G -> H<sup>n</sup>(X;G) -> Tor(H<sup>n+1</sup>(X;Z),G) -> 0
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(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(Z<sup>s</sup>,n)] up the fibration sequence, and the ones on the Tor side are what are left.