Binary forms with the same value set III. The case of ${\mathbf{D}}_3$ and ${\mathbf{D}}_{\mathbf{6}}$

Let $F,G\in \mathbb{Z}[X,Y]$ be binary forms of degree $\ge 3$ with automorphism groups isomorphic to the dihedral group of cardinality $6$ or $12$. We characterize exactly when $F$ and $G$ have the same value set, i.e. $F({\mathbb{Z}}^2)=G({\mathbb{Z}}^2)$.