Counting number fields with symplectic Galois group

Let $n\geq 1$ and let $\ell$ be an odd prime. Let $G=\mathrm{PGSp}_{2n}(\mathbb{F}_\ell)$ or $\mathrm{GSp}_{2n}(\mathbb{F}_\ell)$, and fix a faithful transitive permutation representation $π:G\longrightarrow S_d$. We study degree-$d$ number fields whose Galois closures have Galois group $G$ and whos