Let $G$ be a split reductive group over a nonarchimedean local field $F$ (I'm particularly interested in the case of $\operatorname{GSp}_{2n}$). Given a parahoric subgroup $K \subset G(F)$, and a parabolic subgroup $P$, is there a "nice" group-theoretic description of the orbits of $K$ on the flag variety $\mathcal{F}(F) = P(F) \backslash G(F)$?

If $K$ is contained in a hyperspecial subgroup, which we can take to be $G(\mathcal{O})$ for some reductive integral model of $G$, then $K$ is the inverse image of a parabolic $Q$ in $G(k)$ where $k$ is the residue field. In this case $P(F) G(\mathcal{O}) = G(F)$, so we have $$P(F) \backslash G(F) / K = P(\mathcal{O}) \backslash G(\mathcal{O}) / K = P(k) \backslash G(k) / Q(k) = W_{M_P} \backslash W_G / W_{M_Q},$$ where $M_P$ is the Levi of $P$ and $W_{M_P}$ its Weyl group.

However, this doesn't work for parahorics that aren't contained in a hyperspecial, like the paramodular group in $\operatorname{GSp}_4$. Is there some way of describing the orbits of these subgroups on the flag variety (maybe using affine Weyl groups)?

Let $G$ be a split reductive group over a nonarchimedean local field $F$ (I'm particularly interested in the case of $\operatorname{GSp}_{2n}$).

Given a parahoric subgroup $K \subset G(F)$, and a parabolic subgroup $P$, is there a "nice" group-theoretic description of the orbits of $K$ on the flag variety $\mathcal{F}(F) = P(F) \backslash G(F)$?

I have the following conjecture: if $T$ is a split maximal torus in $G$ and $\widetilde{W} = N_G(T)(F) / T(\mathcal{O}_F)$ is the extended affine Weyl group, then $\widetilde{W}$ is generated by a set $S$ of simple reflections, and there is a bijection between standard parahorics and proper subsets of $S$. I think that if $K_J$ is a standard parahoric corresponding to some $J \subset S$, then $$ P \backslash G / K = W_{M_P} \backslash W_G / \pi(W_J) $$ where $W_J$ is the subgroup of $\widetilde{W}$ generated by $J$, and $\pi(W_J)$ is its image in the usual Weyl group. This conjecture is true if $K$ is contained in $G(\mathcal{O}_F)$. Does anyone know if it is true in general?