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 $Hom(M,k)\times M\rightarrow k$.



Transgression in terms of k-invariant for chain complexes
I think the difficulty is that you are assuming that $X$ only has homology in two degrees, but are then looking at the cohomology spectral sequence. (To get sensible answers I seem to have to take cohomology to be negatively graded.)


Suppose instead that $H^0(X;k)=k$ and $H^{-n}(X;k)=N$ are the only two non-trivial cohomology groups. The inclusion of the constant $0$-cochains gives a map $k \to C^\*(X;k)$ whose mapping cone has homology $N[-n]$, so there is an exact triangle
$$N[-n-1] \overset{\Sigma^{-n-1} \kappa}\to k \to C^\*(X;k)$$
in the derived category of complexes of $kG$-modules, 
defining the $k$-invariant $\kappa \in Ext\_{kG}^{n+1}(N,k)$.


Then $C^\*((EG \times X)/G;k) \cong (C^\*(EG;k) \otimes C^\*(X;k))^G$, and $C^\*(EG;k)$ is a complex of finitely generated free $kG$-modules so $(C^\*(EG;k) \otimes -)^G$ is exact, hence there is an exact triangle
$$(C^\*(EG;k) \otimes N[-n-1])^G \to C^\*(G;k) \to C^\*((EG \times X)/G;k).$$
in the derived category of $k$-modules. The associated long exact sequence
$$H^\*(G;n) \to H^\*((EG \times X)/G;k) \to H^{\*+n}(G;N) \overset{d}\to H^{\*-1}(G;k)$$
is then the cohomology spectral sequence, and the connecting map $d$ is now manifestly given by product with $\kappa$.


