We study the distribution of the traces of the Frobenius endomorphism of genus g curves which are quartic non-cyclic covers of P 1 Fq , 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 P 1 Fq with Galois group isomorphic to r copies of Z 2Z. For r = 1, we recover the already known hyperelliptic case.