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We show that the natural morphism $\phi:\pi_1(X_{\eta},x_{\eta})\to\pi_1(X,x)_{\eta}$ between the fundamental group scheme of the generic fiber$X_{\eta}$ of a scheme $X$ over a connected Dedekind scheme and the genericfiber of the fundamental group scheme of $X$ is always faithfully flat. As anapplication we give a necessary and sufficient condition for a finite,dominated pointed $G$-torsor over $X_{\eta}$ to be extended over $X$. Wefinally provide examples where $\phi:\pi_1(X_{\eta},x_{\eta})\to\pi_1(X,x)_{\eta}$ is an isomorphism..

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