For a Weyl group $W$ and a toric $W$-permutohedral variety $X_P$, there is another toric variety $X_{P/W_K}$ for any parabolic subgroup $W_K$ of $W$ since the quotient $P/W_K$ can be identified with a polytope. We construct an explicit algebra isomorphism between \(H^*({X_{P / W_K}};\mathbb{Q})\) and $H^*(X_P;\mathbb{Q})^{W_K}$. We also generalize this algebra isomorphism to arbitrary finite Coxeter groups. Our work gives positive answers to two open questions of Horiguchi-Masuda-Shareshain-Song.