Let $X$ be a smooth geometrically connected scheme over a field $k$ of characteristic 0 (but not necessarily algebraically closed, I can take it to be a number field). Let $F$ be a finite algebraic group over $k$. 



> 
> Is the following statement true: $H\_{et}^1(X,F) = H^1(\pi\_1^{et}(X), F(\bar{k}))$?
> 
> 
> 



Is $H_{et}^1(X,F) = H^1(\pi_1^{et}(X), F(\bar{k}))$ true?
This is true, yes. More generally, if $X$ is a scheme and $F$ is a locally constant étale sheaf of finite abelian groups on $X$, then
$$
H^1\_{et}(X,F) = H^1(\Pi\_1^{et}(X), \tilde F),
$$
where $\Pi\_1^{et}(X)$ is the étale fundamental pro-groupoid of $X$ and $\tilde F$ a certain local system on $\Pi\_1^{et}(X)$ corresponding to $F$. This follows from Proposition 5.9 in Friedlander's book on étale homotopy [the assumption there that $X$ is locally noetherian is not needed when $F$ is finite], together with the fact that $H^1$ of a (pro-)space is the same as $H^1$ of its 1-truncation.


When $X$ is connected and $x$ is a geometric point of $X$, $\Pi\_1^{et}(X)\simeq B\pi\_1^{et}(X,x)$, so that local systems on $\Pi\_1^{et}(X)$ can be identified with $\pi\_1^{et}(X,x)$-modules. Under this identification, the local system $\tilde F$ is just $F(x)$ with its $\pi\_1^{et}(X,x)$-action.


