  We consider the Galois group $G_2(K)$ of the maximal unramified $2$-extension
of $K$ where $K/\mathbb{Q}$ is cyclic of degree $3$. We also consider the group
$G^+_2(K)$ where ramification is allowed at infinity. In the spirit of the
Cohen-Lenstra heuristics, we identify certain types of pro-$2$ group as the
natural spaces where $G_2(K)$ and $G^+_2(K)$ live when the $2$-class group of
$K$ is $2$-generated. While we do not have a theoretical scheme for assigning
probabilities, we present data and make some observations and conjectures about
the distribution of such groups.
