  Let $f\left(K\right)$ be the number of unramified extensions $L/K$ of a
quadratic number field $K$ with $\mathrm{Gal}\left(L/K\right)=H$ and
$\mathrm{Gal}\left(L/\mathbb{Q}\right)=G$ where $G$ is a central extension of
$\mathbb{F}_{2}^{n}$ by $\mathbb{F}_{2}$. We find a function $g\left(K\right)$
such that $f/g$ has finite moments and a distribution on its values. We show
this distribution is a point mass when $H$ is non-abelian and the Cohen-Lenstra
distribution when $H$ is abelian, despite the fact that the set of values of
$f/g$ do not form a discrete set. We prove an explicit formula for $f$ as well
as a refined counting function with local conditions. We also determine
correlations of such counting functions for different groups $G$. Lastly we
formulate a conjecture about moments and correlations for any pair of 2-groups
$\left(G,H\right)$.
