Let $p:X\rightarrow Y$ be a double cover of curves, denote by $$SU_n:=(p_*SL_n(\mathcal O_X))^{\tilde{\sigma}}$$ i.e. the $\tilde{\sigma}-$invariant part, the action of $\tilde{\sigma}$ is given by $$\tilde{\sigma}(g)=\,^t(g\circ\sigma)^{-1}$$ where $\sigma$ is the involution induced by the double cover. $SU_n$ is well knowing to be a parahoric group shceme in the sens of Bruhat-Tits.

My question: What is $\pi_1(({SU_n})_\eta)$ ? (the algebraic fundamental groupe). where $\eta$ is the generic point of $Y$.

Thanks