Distributions of unramified 2-group extensions of quadratic fields

This is a continuation of the talk by Brandon Alberts. We will discuss the Cohen-Lenstra heuristics from the point of view of counting unramified number field extensions of quadratic fields. We will apply the field counting results discussed in the previous talk to prove distributional results.

In particular we determine the distribution of the function counting extensions with Galois groups which are central extensions of $(F_2)^n$ by $F2$. 
We show it is either a point mass or the Cohen-Lenstra distribution despite the fact that the set of values do not form a discrete set.
Which distribution occurs depends on the structure of a graph formed from the generators of G, and in particular whether or not it is complete
bipartite. We will also put forth a conjecture about asymptotics and distributions of such extensions for G a general 2-group.