Let $P \subset \mathbb R^n$ be a compact rational polytope (the affine space spanned by $P$ may not be $\mathbb R^n$). Let $G \subset {\bf GL}(n,\mathbb Z)$ be an arithmetic group. Let $$C = {\rm Conv}(\bigcup_{g\in G} \ g\cdot P)$$ be the convex hull.

Question: does there exist a compact rational polytope $Q \subset C$ such that $\bigcup_{g\in G}\ g\cdot G = C$ ?