I am trying to understand constructions of exceptional groups of type $G_2$ (over rings). In this post, by a model (of type $G_2$) I mean an affine smooth group scheme over $\mathbb{Z}$ such that the fibres are connected simple algebraic groups of type $G_2$.

In Gross' paper Groups over Z (see Page~272) a model $\mathbb{G}$ is given using Coxeter's integral octonions, and it is mentioned there that $\mathbb{G}(\mathbb{Z})$ is isomorphic to $G_2(\mathbb{F}_2)$ as abstract groups. Models are not unique: For example, this $\mathbb{G}$ is a non-split model, and there is also a split one as given in Appendix~B of Conrad's paper Non-split reductive groups over Z.

Question: Do we have $\mathbb{G}'(\mathbb{Z})\cong G_2(\mathbb{F}_2)$ for any model $\mathbb{G}'$ of type $G_2$?

I am trying to understand constructions of exceptional groups of type $G_2$ (over rings). In this post, by a model (of type $G_2$) I mean an affine smooth group scheme over $\mathbb{Z}$ such that the fibres are connected simple algebraic groups of type $G_2$.

In Gross' paper Groups over Z (see Page 272) a model $\mathbb{G}$ is given using Coxeter's integral octonions, and it is mentioned there that $\mathbb{G}(\mathbb{Z})$ is isomorphic to $G_2(\mathbb{F}_2)$ as abstract groups. Models are not unique: For example, this $\mathbb{G}$ is a non-split model, and there is also a split one as given in Appendix B of Conrad's paper Non-split reductive groups over Z.

Question: Do we have $\mathbb{G}'(\mathbb{Z})\cong G_2(\mathbb{F}_2)$ for every model $\mathbb{G}'$ of type $G_2$?