I am looking for a reference for the following. Say we have a $G$-space $X$ whose homology groups (in field coefficients $k$) are non-zero only in dimension zero and for a fixed $n>0$. Let $M$ denote the top homology group of $X$. 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 $Ext^{n+1}_{kG}(k,M)$. I believe the product looks like $Ext^{0}_{kG}(k,Hom(M,k))\times Ext^{n+1}_{kG}(k,M)\rightarrow Ext^{n+1}(k,k)$ induced by the pairing $Hom(M,k)\times M\rightarrow k$.