  Let $G$ be a transitive subgroup of $S_d$ and $E$ be a CM field of degree
$2d$ with a maximal totally real $G$-field. If the Galois group of the Galois
closure of $E$ is isomorphic to the wreath product of $C_2$ and $G$, then we
say that $E$ is a $G$-Weyl CM field.
  Let $N_{2d}^{\textrm{Weyl}}(X,G)$ count the $G$-Weyl CM fields $E$ of degree
$2d$ with discriminant $|d_E| \leq X$ and define \begin{align*}
N_{2d}^{\textrm{Weyl}}(X):=\sum_{G \leq S_d}N_{2d}^{\textrm{Weyl}}(X,G).
\end{align*} Further, let $N_{2d}^{\textrm{cm}}(X)$ count the CM fields $E$ of
degree $2d$ with discriminant $|d_E| \leq X$. Assuming a weak form of the upper
bound in Malle's conjecture which is known to be true in many cases, we build
upon an approach of Kl\"uners to prove that \begin{align*}
\frac{N_{2d}^{\textrm{Weyl}}(X,G)}{N_{2d}^{\textrm{cm}}(X)} = C(d, G) +
O(X^{-\alpha(d,G)}) \end{align*} and \begin{align}
\frac{N_{2d}^{\textrm{Weyl}}(X)}{N_{2d}^{\textrm{cm}}(X)} = 1 +
O(X^{-\beta(d)}) \qquad \qquad (0.1) \end{align} for some explicit positive
constants $C(d,G), \alpha(d,G)$, and $\beta(d)$.
  We then apply these distribution results to study the Colmez conjecture.
Using the recently proved averaged Colmez conjecture, we deduce that the Colmez
conjecture is true for $G$-Weyl CM fields. Combined with (0.1), we conclude
that the Colmez conjecture is true for an asymptotic density of 100% of CM
fields of degree $2d$; in other words, the Colmez conjecture is true for a
random CM field.
