Let $G$ be a reductive group defined over $\mathbb{Z}_{p}$ and let $H$ be a closed reductive subgroup of $G$. Let $Q_{G}$ be a parabolic subgroup of $G$ with Levi decomposition $Q_{G} = L_{G} \ltimes N_{G}$ and let $Q_{H}$ be a parabolic subgroup of $H$ with Levi decomposition $Q_{H} = L_{H} \ltimes N_{H}$. Let $M_{H}$ be a mirabolic subgroup of $Q_{H}$ i.e. there is a normal subgroup $L_{M} \subset L_{H}$ such that $M_{H} = L_{H} \ltimes N_{H}$.
Suppose that the pair $(G, M_{H})$ is spherical i.e. $M_{H}$ has an open orbit on the (parabolic) flag variety $\mathcal{F}_{G} := G/\bar{Q}_{G}$ where $\bar{Q}_{G}$ is the conjugate of $Q_{G}$ under the long Weyl element. Suppose that this orbit contains the image of $1$ for simplicity.
Is it the case that the $M_{H}$-stabiliser of the image of $1$ in $\mathcal{F}_{G}$ is compatible with the Levi decomposition of $\bar{Q}_{G}$ i.e. $M_{H} \cap \bar{Q}_{G}$, decomposes as a product $(M_{H} \cap L_{G}) \ltimes (M_{H} \cap N_{G})$? If not, are there conditions on $G, H$ which would make this true?
