Let $H^\circ\subseteq H$ be the connected component of unity and let $M^\circ$ be weight lattice of $X^\circ:=G/H^\circ$. Since $\pi_0(H)=H/H^\circ$ acts as automorphisms on $X^\circ$ it follows that $\pi_0(H)$ is the dual group of, hence isomorphic to, the finite abelian group $M^\circ/M$. Thus, everything boils down to compute $M^\circ$. Now, this has been done by Hofscheier in his paper Containment Relations among Spherical Subgroups. Corollary 1.3 states, in particular, that $$M^\circ=\{\chi\in M_{\mathbb Q}\cap\mathcal X(B)\mid \langle\rho(\mathcal D),\chi\rangle\subseteq\mathbb Z\}.$$

Let $H^\circ\subseteq H$ be the connected component of unity and let $M^\circ$ be weight lattice of $X^\circ:=G/H^\circ$. Since $\pi_0(H)=H/H^\circ$ acts as automorphisms on $X^\circ$ it follows that $\pi_0(H)$ is the dual group of, hence isomorphic to, the finite abelian group $M^\circ/M$. Thus, everything boils down to compute $M^\circ$. Now, this has been done by Hofscheier in his paper Containment Relations among Spherical Subgroups. Corollary 1.5 states, in particular, that $$M^\circ=\{\chi\in M_{\mathbb Q}\cap\mathcal X(B)\mid \langle\rho(\mathcal D),\chi\rangle\subseteq\mathbb Z\}.$$