Arithmetically equivalent siblings (awaiting review)

If $K$ is a degree $n$ number field whose Galois closure has Galois group $G\leq S_n$, then $K$ may have arithmetically equivalent fields. The number of these is a function of $G$.

Inside $G$, we have $H_1$, the stabilizer subgroup of $1$. By the Galois correspondence, an arithmetically equivalent field corresponds to a conjugacy class of subgroups with representative $H_2$ such that $$ |H_1 \cap C| = |H_2\cap C|$$ for every conjugacy class $C$ of $G$.