We know that the Borel group cohomology (group cohomology of measurable functions) of a group $G$, ${\cal H}_B^d(G,Z)$, is given by the cohomology of the classifying space: ${\cal H}_B^d(G,Z)=H^d(BG,Z)$. However, ${\cal H}_B^d(G,R/Z)\neq H^d(BG,R/Z)$, since for example: $H^d(BU(1),R/Z) = R/Z$ for even $d$ and $H^d(BU(1),R/Z) = 0$ for odd $d$; while ${\cal H}_B^d(BU(1),R/Z)=0$ for even $d$ and ${\cal H}_B^d(BU(1),R/Z)=Z$ for odd $d$. Instead, we have ${\cal H}^d(G,R/Z) = H^{d+1}(BG,Z)$.

My question is that do we have any relations between ${\cal H}_B^*(G,M)$ and $H^*(BG,M')$ where $G$ can be continuous, $M$ can be $Z_n$, and $M'$ can be different from $M$? In particular, I would like to know how ${\cal H}_B^*(G,Z_n)$ is related to $H^*(BG,M')$ when $G$ is continuous.

A related question group cohomology and cohomology of classifying space is closed. I hope it can be reopen.