I am looking for a reference for the following. Say we have a $G$-space $X$ whose homology groups are non-zero only in dimension zero and for a fixed $n>0$. Then in the cohomology spectral sequence for the fibration $X\rightarrow (EG\times X)/G\rightarrow BG$ the transgression is multiplication by the $k$-invariant of $M$. The $k$-invariant lives in $Ext^{n+1}_{\mathbb{Z}G}(k,M)$.