  Let $F, G \in \mathbb{Z}[X, Y]$ be binary forms of degree $\geq 3$, 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 ${\rm GL}(2,
\mathbb{Z})$--equivalent.
