Wood's conjecture on unramified extensions of quadratic fields

Let $G'$ be a finite group and $G$ a subgroup. Write $\mathbf{E}^{\pm}(G',G)$ for the average number of unramified extensions $L/K$ of quadratic fields $K$, unramified at finite primes, such that $\operatorname{Gal}(L/K)=G$ and $\operatorname{Gal}(L/\mathbb{Q})=G'$. Let $c$ be a conjugacy class of elements of order $2$ in $G'$, and let $H_{2}(G',c)$ and $\operatorname{Aut}_{G'}(G)$ denote the corresponding reduced Schur multiplier and group of $G'$-equivariant automorphisms. The parameter $u$ is $0$ in the imaginary case and $1$ in the real case.

Wood's conjecture. If there is a unique conjugacy class $c$ of order $2$ elements of $G'$ not contained in $G$, then

$$\mathbf{E}^{\pm}(G',G)=\frac{\left|H_{2}(G',c)[2]\right|}{\left|c\right|^{u}\left|\operatorname{Aut}_{G'}(G)\right|}.$$

Otherwise,

$$\mathbf{E}^{\pm}(G',G)=\infty.$$

This conjecture generalizes the Cohen–Lenstra question from abelian class groups to non-abelian unramified extensions; Wood proved supporting results over function fields, while the stated number-field distribution remains open.