  We construct here many families of K3 surfaces that one can obtain as
quotients of algebraic surfaces by some subgroups of the rank four complex
reflection groups. We find in total 15 families with at worst
$ADE$--singularities. In particular we classify all the K3 surfaces that can be
obtained as quotients by the derived subgroup of the previous complex
reflection groups. We prove our results by using the geometry of the weighted
projective spaces where these surfaces are embedded and the theory of Springer
and Lehrer-Springer on properties of complex reflection groups. This
construction generalizes a previous construction by W. Barth and the second
author.
