Let $X$ be a connected cell complex with fundamental group $G$ and $(n-1)$-connected universal covering space. Let $\Pi=\pi_n(X)$. We may construct a $K(G,1)$ complex $K$ by adjoining cells of dimension $>n$ to $X$. Let $c_X:X\to{K}$ be the inclusion. If we view $c_X$ as a fibration, the $k$-invariant $k_1(X)$ is the first obstruction to the existence of a section for $c_X$.

Question: Can $k_1(X)$ be identified with the extension class of the iterated extension $0\to\Pi\to{C_n/dC_{n+1}}\to\dots\to{C_0}\to\mathbb{Z}\to0$, as elements of $H^{n+1}(G;\Pi)$?

This may well be somewhere in the papers of Eilenberg, Mac Lane and/or JHC Whitehead from the late 1940s, but I had not been able to track down a published proof. (The Homotopy Addition Theorem may be used to identify the homological extension class with the first obstruction to retracting $K$ onto $X$, but that is not quite the same problem.)