Is there a theorem to estimate the number of integer polynomials with Frobenius class in $C$, where $C$ is a subset $G=S_n$ that is stable under conjugation? In other words, For a fixed prime $p$,
$\#\{ f(x) \in \mathbb{Z}[x]: \{\text{Frob}_{p}\} \subseteq C\} \ll$ ?
where $\{\text{Frob}_{p}\}$ is a conjugacy class in $S_n$, i.e., Frobenius class.