  We study the distribution of the traces of the Frobenius endomorphism of
genus $g$ curves which are quartic non-cyclic covers of
$\mathbb{P}^{1}_{\mathbb{F}_{q}}$, as the curve varies in an irreducible
component of the moduli space. We show that for $q$ fixed, the limiting
distribution of the trace of Frobenius equals the sum of $q + 1$ independent
random discrete variables. We also show that when both $g$ and $q$ go to
infinity, the normalized trace has a standard complex Gaussian distribution.
Finally, we extend these computations to the general case of arbitrary covers
of $\mathbb{P}^{1}_{\mathbb{F}_{q}}$ with Galois group isomorphic to $r$ copies
of $\mathbb{Z}/2\mathbb{Z}$. For $r = 1$, we recover the already known
hyperelliptic case. We also include an appendix by Alina Bucur giving the
heuristic of these distributions.
