  Let $R$ be a discrete valuation ring with fraction field $K$. Let $X$ be a
proper and faithfully flat $R$-scheme, endowed with a section $x \in X(R)$,
with connected and reduced generic fibre $X_{\eta}$. Let $f: Y \rightarrow
X_{\eta}$ be a finite Nori-reduced $G$-torsor. In this paper we provide a
useful criterion to extend $f: Y \rightarrow X_{\eta}$ to a torsor over $X$.
Furthermore in the particular situation where $R$ is a complete discrete
valuation ring of residue characteristic $p>0$ and $X\to \text{Spec}(R)$ is
smooth we apply our criterion to prove that the natural morphism $\psi^{(p')}:
\pi(X_{\eta},x_{\eta})^{(p')}\to \pi(X,x)_{\eta}^{(p')}$ between the
prime-to-$p$ fundamental group scheme of $X_{\eta}$ and the generic fibre of
the prime-to-$p$ fundamental group scheme of $X$ is an isomorphism. This
generalizes a well known result for the \'etale fundamental group. The methods
used are purely tannakian.
