Let $F(x,y)$ be an irreducible binary form with integer coefficients and degree $d \geq 3$. We say that an integer $n$ is primitively represented by $F$ if there exist coprime integers $(x, y)$ such that $F(x,y)=n$. We also say that a subset $\mathcal S \subset \mathbb Z$ is primitively represented by $F$ if every $n \in \mathcal S$ is primitively represented by $F$.

Next, for any matrix

$$A = \left(\begin{matrix}a & b\\c & d\end{matrix}\right)$$

with rational entries, define

$$F_A(x,y) = F(ax + by, cx + dy).$$

We say that two binary forms $F$ and $G$ are equivalent if there exists $A \in \operatorname{GL}_2(\mathbb Z)$ such that $F_A = G$.

I am interested in the following question: given two distinct non-equivalent irreducible binary forms $F$ and $G$ of the same degree $d \geq 3$ and discriminant, when is it possible for them to represent primitively the same infinite subset of the integers?

It has been pointed out to me by @StanleyYaoXiao that from a given binary form $F$ it is possible to construct $G$ satisfying the above criteria whenever $F$ has a non-trivial $\mathbb Q$-automorphism (i.e. $A \in \operatorname{GL}_2(\mathbb Q) \setminus \operatorname{GL}_2(\mathbb Z)$ such that $F_A = F$). Are there any other instances or examples when this can happen?