Let

$$1 \longrightarrow K \longrightarrow G \longrightarrow Q \longrightarrow 1$$

be a short exact sequence of groups and let $M$ be a $\mathbb{Z}[G]$-module. The Hochschild--Serre spectral sequence in homology is of the form

$$E^2_{pq} = H_p(Q;H_q(K;M)) \Longrightarrow H_{p+q}(G;M).$$

All the sources I've consulted construct this via a Grothendieck spectral sequence. For something I am doing, what would be great is if there is a free resolution $R_{\bullet} \rightarrow \mathbb{Z}$ of the trivial $\mathbb{Z}[G]$-module $\mathbb{Z}$ and a filtration $\mathcal{F}_{\bullet} R_{\bullet}$ of $R_{\bullet}$ such that for all $\mathbb{Z}[G]$-modules $M$, the above spectral sequence is the one associated to the filtration

$$\mathcal{G}_k(R_{\bullet} \otimes M) = (\mathcal{F}_k R_{\bullet}) \otimes M$$ of the chain complex $R_{\bullet} \otimes M$ computing $H_p(G;M)$. It would be even better if this filtration resolution was reasonably explicit, though I could get by with something non-explicit if I had to. Does anyone know where I could find this?