Let $k = \overline{\mathbb F_p}$ and let $G$ be an affine finite type group scheme over $\mathbb Z$. Suppose that $T$ is a $G_k$-torsor over $k$.

Does $T$ lift to characteristic zero (in the weakest possible sense)?

That is, does there exist a complete local (regular) ring $R$ with residue field $k$ such that

1. $R$ is of characteristic zero, and

2. 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 $H^1_{fppf}(R,G_R)\to H^1_{fppf}(k,G_k)$ is surjective for all complete local regular rings $R$ with residue field $k$?

What if $G$ is smooth and connected over $\mathbb Z$?