Finite subgroups of $mathrm{PGL}_2(mathbb{Q})$ and number fields with large class groups

For each finite subgroup $G$ of $\mathrm{PGL}_2(\mathbb{Q})$, and for each integer $n$ coprime to $6$, we construct explicitly infinitely many Galois extensions of $\mathbb{Q}$ with group $G$ and whose ideal class group has $n$-rank at least #G-1. This gives new $n$-rank records for class groups of number fields.