Universal coefficient theorem allows us to calculate 
$H^\*(X,M)$ 
from $H\_\*(X,Z)$.
Do we have a "dual" universal coefficient theorem that allows us to calculate $H\_\*(X,M)$ from $H^\*(X,Z)$? 


Here $Z$ is the set of integers.



A "dual" universal coefficient theorem
Yes, there is such a universal coefficient theorem. 


$$0 \to Ext(H^{q+1}(X,R), G) \to H\_q(X, G) \to Hom(H^q(X, R), G) \to 0$$


see Theorem 6.5.12 in Spanier's textbook "Algebraic Topology". It's on page 248. 


