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Let $X$ be a scheme with base point $x:\operatorname{Spec}(K)\rightarrow X$ and algebraic fundamental group $\pi_1(X, x)$. Let $H$ be a normal subgroup of $\pi_1(X,x)$. How do we construct a scheme $X_H$ and a morphism $p:X_H\rightarrow X$ such that $p_{*}(\pi_1(X_H))=H$ where $p_{*}$ is the induced map on fundamental groups?

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    The $\pi_1$ that I have in mind is the one in SGA1. Sorry for the confusion.2011-09-13

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