On the étale site of a scheme $X$, there is a spectral sequence associated to the data of an étale hypercover $K$ of $X$ and an abelian étale sheaf $\mathcal F$ on $X$: $$E_2^{p,q}=\check H^p(K,\underline H^q(\mathcal F))\Rightarrow H^{p+q}(X,\mathcal F).$$ (See https://stacks.math.columbia.edu/tag/01GY.) Here $\underline H^q(\mathcal F)$ is the étale presheaf with value $\underline H^q(\mathcal F)(U)=H^q(U,\mathcal F)$ on $U\to X$ étale.
Does the same spectral sequence exist for $\ell$-adic cohomology ($\mathbf{Z}_\ell$ or $\mathbf{Q}_\ell$ sheaves), with $\underline H^q(\mathcal F)$ a presheaf of $\mathbf{Z}_\ell$-modules or $\mathbf{Q}_\ell$-vector spaces? (With sufficient finiteness hypotheses on $X$ such as being of finite type over a field.) If so, is there a reference? (The closest I have been able to find to something mentioning this are the notes of Brian Conrad titled ‘Cohomological Descent.’)
Thank you.