Counting $C_2\wr S_4$ fields with a power saving error term

Let $N_d(G,X)$ denote the number of degree $d$ extensions of $\mathbb{Q}$ with Galois closure $G$ and $\mathrm{\mid }{\mathrm{\Delta }}_K\mathrm{\mid }\le X$. Malle’s conjecture predicts an asymptotic of the form $N_d(G,X)\sim C{X}^{\alpha }(\log X{)}^{\beta }$. Previously, Klüners proved Malle’s conjecture for $G=C_2\wr S_4$. His proof gives a power savings of $O(X^{7\mathrm{/}8})$. We improve Klüners’ result by establishing a stronger power saving error term for the count of such fields. Specifically, we show $N_8(C_2\wr S_4,X)=CX+O(X^{3\mathrm{/}4-1\mathrm{/}30})$. Additionally, we obtain new bounds on $N_8(G,X)$ for the groups $S_4$, $C_2^3⋊S_4$, $G{L}_2({\mathbb{F}}_3)$, and $Q_8⋊S_4$ as permutation subgroups of $S_8$.