We show that the natural morphism ϕ:π1(Xη,xη)→π1(X,x)η between the fundamental group scheme of the generic fiber
Xη 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 Xη to be extended over X. We
finally provide examples where ϕ:π1(Xη,xη)→π1(X,x)η is an isomorphism..Comment: 19 pages, final versio