Let $G$ be a reductive group over a local non-arch field $F$. For convenience let's assume $G$ has anisotropic center (or even that $G$ is semisimple if preferred). Let $x$ be any point in the building $\mathcal{B}(G,F)$, and $G_x$ its stabilizer in $G(F)$. It also contains the parahoric, which I denote by $G_{x,0}$, attached to (the facet that contains) $x$.
For the purpose of some computation of orbital integrals, I'm interested in this quotient group $G_x/G_{x,0}$. When $x$ is a vertex or a general point of a facet, 3.5 of Tits' Corvallis article "Reductive groups over local fields" provide an algorithm: Let $G'$ be the centralizer of a maximal split torus. Then $G'(F)$ maps to the automorphism group of the affine Dynkin diagram. The kernel of this map can be computed - let's just call it $\text{ker}$. Tits explains how $G_x/G_{x,0}$ appears as a subgroup of $G'(F)/\text{ker}$ (more precisely, he explains this in the group scheme language, and the result I mention can be deduced).
I'm wondering if exactly the same result is true (and moreover, does it always come from an explicit isomorphism from a subgroup of $G'(F)/\text{ker}$ to $G_x/G_{x,0}$) when $x$ is any point or even when $x$ is replaced by any bounded subset of an apartment. Thank you!
p.s. I fail to find such a statement in the original paper of Bruhat-Tits. But I'm very unfamiliar with that paper - hope I'm not being stupid.