Let $X$ be a variety over a field $k$ with separable closure $k_s$. Let $A$, $B$ be étale sheaves on $X$. Consider now the Hochschild–Serre spectral sequences.

\begin{align*} E_2^{pq}: H^p(k, H^q(X_{k_s}, A)) & {}\Rightarrow H^{p + q}(X, A) \\ E_2^{pq}: H^p(k, H^q(X_{k_s}, B)) & {}\Rightarrow H^{p + q}(X, B) \\ E_2^{pq}: H^p(k, H^q(X_{k_s}, A \otimes B)) & {}\Rightarrow H^{p + q}(X, A \otimes B). \end{align*}

The cup product for $H^{\bullet}(X_{k_s}, \cdot)$ gives a bilinear pairing $H^q(X_{k_s}, A) \times H^{q'}(X_{k_s}, B) \to H^{q + q'}(X_{k_s}, A \otimes B)$. This induces via the cup product on $H^{\bullet}(k, \cdot)$ a bilinear pairing $E^{pq}_2 \times E^{p' q'}_2 \to E^{(p + p')(q + q')}_2$. This should be the same bilinear pairing (up to a sign) as the one coming from the usual cup product pairing $H^{p + q}(X, A) \times H^{p' + q'}(X, B) \to H^{p + p' + q +q'}(X, A \otimes C)$ (part of the statement being that the cup product is compatible with the filtration).

For the Serre spectral sequence in algebraic topology this is described in Hatcher - Algebraic topology - Spectral sequences page 543 or, Hutchings - Introduction to spectral sequences. Does anyone know of a good reference where this is explained for the Hochschild–Serre spectral sequence? The Leray spectral sequence would also be fine, since Hochschild–Serre is a special case. Especially the signs seems like they might be subtle.