I am looking for answers/references for the following constructions/statements.
Let $X$ be a scheme over a field $K$, with geometric etale fundamental group $\pi_1(\overline{X},\overline{b})$.
- There exists a classifying space:
$B_{et}\pi_1(\overline{X},\overline{b})$, with the property that for every finite dimensional $\pi_1(\overline{X},\overline{b})$ representation $V$, one has: $$H^n(\pi_1(\overline{X},\overline{b}), V) \cong H^n_{et}(B_{et}\pi_1(\overline{X},\overline{b}), \mathbb{V}).$$
Here $\mathbb{V}$ is the etale local system on $B_{et}\pi_1(\overline{X},\overline{b})$ that corresponds to $V$, and $H^n(\pi_1(\overline{X},\overline{b}), V)$ is the $n$-th group cohomology of $\pi_1(\overline{X},\overline{b})$ with coefficients in $V$.
Denote by $\tilde{X}$ the projective limit of finite etale covers of $X$, then the etale homotopy fibre of the map $X\longrightarrow B_{et}\pi_1(\overline{X},\overline{b})$ is $\tilde{X}$.
Is there a Serre spectral sequence for the etale homotopy fibration $$\tilde{X}\longrightarrow X\longrightarrow B_{et}\pi_1(\overline{X},\overline{b})?$$