Cohen–Lenstra–Martinet average torsion conjecture for class groups of G-extensions

Let $k$ be a number field, let $G$ be a transitive permutation group, and let $p$ be a prime such that $p\nmid \lvert G\rvert$. For a real number $X>1$, let $E_k(G,X)$ be the set of $G$-extensions $K$ of $k$ with $\lvert\operatorname{Disc}(K)\rvert\leq X$, and let $h_p(K)$ denote the size of the $p$-torsion subgroup of $\operatorname{Cl}(K)$. Cohen–Lenstra–Martinet conjecture. There exists a constant $c_{k,G,p}>0$ such that

$$\lim_{X\to\infty}\frac{1}{\lvert E_k(G,X)\rvert}\sum_{K\in E_k(G,X)}h_p(K)=c_{k,G,p}.$$

This conjecture predicts that the average size of $p$-torsion in class groups of $G$-extensions is finite, with the constant determined by $k$, $G$, and $p$. The paper proves the conjecture for $p=3$ and groups of the form $C_2\wr H$ when $H$ is any nilpotent group, extending earlier work for a broad family of permutation groups.