Comparison between the fundamental group scheme of a relative scheme and that of its generic fiber

We show that the natural morphism $ϕ:π_1(X_η,x_η)\to π_1(X,x)_η$ between the fundamental group scheme of the generic fiber $X_η$ of a scheme $X$ over a connected Dedekind scheme and the generic fiber of the fundamental group scheme of $X$ is always faithfully flat. As an application we give a necessary and sufficient condition for a finite, dominated pointed $G$-torsor over $X_η$ to be extended over $X$. We finally provide examples where $ϕ:π_1(X_η,x_η)\to π_1(X,x)_η$ is an isomorphism..

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