Lifting torsors in characteristic $p$ to characteristic zero

Question

Let $R$ be a local integral domain with residue field $k$ such that $R$ is of characteristic zero and $k$ is of characteristic $p>0$. Let $G$ be a smooth finite type affine group scheme with geometrically connected fibres over $R$. Suppose that $T$ is a $G_k$-torsor over $k$.

Does $T$ lift to $R$?

That is, does there exist a $G$-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 $H^1_{et}(R,G_R)\to H^1_{et}(k,G_k)$ is surjective for all complete local regular rings $R$ with residue field $k$?

Motivation: I'm trying to understand what it means for a variety over $k$ to lift to characteristic zero. I know curves, ppav's, K3 surfaces and hypersurfaces 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 (under affine group schemes). I hope the comments and answers will shed some light on this matter for me.

Answer 1

The answer is affirmative with $R$ any henselian local ring and $G$ any smooth $R$-group scheme; this is a special case of Lemma 11.4 in Grothendieck's article "Le Groupe de Brauer III: Examples et Complements" (whose proof simplifies quite a bit when $G$ is affine). This is exactly the statement that $\rho_G:{\rm{H}}^1(R,G) \rightarrow {\rm{H}}^1(k, G)$ is surjective in such cases since such torsors split over finite etale covers of the base; strictly speaking that surjectivity result is for commutative $G$ (for which Grothendieck gives a result for higher cohomology too), but the method applies for H$^1$ without a commutativity assumption, as noted in Remark 11.8(3) of loc. cit. (The key insight of Grothendieck is to prove smoothness for the differential from a "scheme of 0-cochains" to a "scheme of 1-cocycles".) It also gives bijectivity.