Normalizer of Twisted Chevalley Groups over Commutative Rings

Let $R$ be a commutative ring with unity. Consider the twisted Chevalley group $G_{\pi ,\sigma }(\mathrm{\Phi },R)$ of type $\phi$ over $R$ and its elementary subgroup $E_{\pi ,\sigma }^{\mathrm{\prime }}(\mathrm{\Phi },R)$. This paper investigates the normalizers of $E_{\pi ,\sigma }^{\mathrm{\prime }}(\mathrm{\Phi },R)$ and $G_{\pi ,\sigma }(\mathrm{\Phi },R)$ in the larger group $G_{\pi ,\sigma }(\mathrm{\Phi },S)$, where $S$ is an extension ring of $R$. We establish that under certain conditions on $R$ these normalizers coincide. Moreover, in the case of adjoint type groups, we show that they are precisely equal to $G_{\pi ,\sigma }(\mathrm{\Phi },R)$.