Extension of torsors and prime to $p$ fundamental group scheme

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 $f: Y \rightarrow X_{\eta}$beafiniteNori-reduced$G$-torsor. In this paper we provide a useful criterion to extend $f:Y\to X_{\eta }$ 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.