Modified Cohen–Lenstra-type distribution conjecture for relative class groups

Let $\Sigma=(G,K_0)$ be the number-field extension data under consideration, let $p$ be a prime, and let $H$ be a finite $p$-group of $p$-rank $r$. Assume that $p$ does not divide the permutation degree of $G$, and that $K_0$ contains the $p$th but not the $p^2$th roots of unity. Let $u=u(\Sigma)$ be the parameter associated with $\Sigma$, and let $\mathcal{K}(\Sigma)$ denote the corresponding family of fields.

Modified class-group distribution conjecture. A given finite $p$-group $H$ occurs as the Sylow $p$-subgroup of a relative class group $\operatorname{Cl}(K/K_0)$ for $K\in\mathcal{K}(\Sigma)$ with probability

$$\nc\, \frac{\prod_{i=1}^{r+u}(p^i-1)}{p^{r(u+1)}}\cdot \frac{1}{|H|^u\,|\operatorname{Aut}(H)|},$$

\nwhere

$$\nc=\frac{1}{\prod_{i=u+1}^\infty(1+p^{-i})}=\frac{(p^2)_u(p)_\infty}{(p)_u(p^2)_\infty}.$$

This modifies the preceding proposed distribution in the presence of $p$th roots of unity, while retaining the Cohen–Lenstra-type dependence on $|H|$ and $|\operatorname{Aut}(H)|$. The parser supplies no resolution evidence, so the conjecture is recorded as open.