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Let $B$ be a path connected topological space with universal cover $p:X \rightarrow B$. Let $G = \pi_1B$, then for a $G$-module $M$ we can form $X \times_G M = X \times M / (g x,m) \sim (x, g m)$ where $M$ is given the discrete topology.

The sections of the bundle $X \times_G M \rightarrow B$ form a locally constant sheaf associated to $M$.

Is there an analogous construction of a sheaf over a scheme $B$ using some sort of action of the etale fundamental group to form a scheme over $B$?

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