Binary forms with the same value set II. The case of ${\mathbf{D}}_4$

Let $F,G\in \mathbb{Z}[X,Y]$ be binary forms of degree $\ge 3$, non-zero discriminant and with automorphism group isomorphic to $D_4$. If $F({\mathbb{Z}}^2)=G({\mathbb{Z}}^2)$, we show that $F$ and $G$ are $\mathrm{G}\mathrm{L}(2,\mathbb{Z})$–equivalent.