Let $F$ be a number field, $p\in\mathbb{Z}$ a prime which is unramified in $F$ and $G$ a connected reductive group over $F$. Moreover $G$ is supposed to be quasi-split over $p$.

Does there exist a finite set of primes $S\subset\mathbb{Z}$ which does not contain $p$ and a split reductive group $\mathcal{G}$ defined over $\mathbb{Z}[1/S]$ such that for some finite extension $K/F$ which is not ramified over $p$ we have the following isomorphism:

$$ \mathcal{G}\times_{\mathbb{Z}[1/S]}K\cong G\times_FK.$$

More generally, given a reductive group $G/F$, does there exist some sort of split integral model $\mathcal{G}$ defined over rings like $\mathbb{Z}$, $\mathbb{Z}[1/S]$ or $\mathcal{O}_F$?