Let $k = \overline{\mathbb F_p}$$R$ be a local integral domain with residue field $k$ such that $R$ is of characteristic zero and let$k$ is of characteristic $p>0$. Let $G$ be an affinea smooth finite type affine group scheme with geometrically connected fibres over $\mathbb Z$$R$. Suppose that $T$ is a $G_k$-torsor over $k$.
Does $T$ lift to characteristic zero (in the weakest possible sense)$R$?
That is, does does there exist a complete local (regular) ring$G$-torsor $R$ with residue field$\mathcal T$ over $k$$R$ such that $\mathcal T \otimes_R k$ is isomorphic to $T$ over $k$?
$R$ is of characteristic zero, and
there is a $G_R$-torsor $\mathcal T$ over $R$ such that $\mathcal T \otimes_R k$ is isomorphic to $T$ over $k$?
If so, can we then show the stronger assertion that the map on cohomology sets $H^1_{fppf}(R,G_R)\to H^1_{fppf}(k,G_k)$$H^1_{et}(R,G_R)\to H^1_{et}(k,G_k)$ is surjective for all complete local regular rings $R$ with residue field $k$?
What ifMotivation: I'm trying to understand what it means for a variety over $G$ is smooth$k$ to lift to characteristic zero. I know curves, ppav's, K3 surfaces and connected overhypersurfaces lift to characteristic zero. I also know examples of Fano varieties, Calabi-Yau threefolds and some surfaces of general type which don't lift to characteristic zero. Out of curiosity I wondered what one can say about liftability of torsors $\mathbb Z$?(under affine group schemes). I hope the comments and answers will shed some light on this matter for me.