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We show that the natural morphism $\varphi :{\pi }_1(X_{\eta },x_{\eta })\to {\pi }_1{(X,x)}_{\eta }$ between the fundamental group scheme of the generic fiber $X_{\eta }$ 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 $X_{\eta }$ to be extended over $X$. We finally provide examples where $\varphi :{\pi }_1(X_{\eta },x_{\eta })\to {\pi }_1{(X,x)}_{\eta }$ is an isomorphism.