  We consider K3 surfaces which are double cover of rational elliptic surfaces.
The former are endowed with a natural elliptic fibration, which is induced by
the latter. There are also other elliptic fibrations on such K3 surfaces, which
are necessarily induced by special linear systems on the rational elliptic
surfaces. We describe these linear systems. In particular, we observe that
every conic bundle on the rational surface induces a genus 1 fibration on the
K3 surface and we classify the singular fibers of the genus 1 fibration on the
K3 surface it terms of singular fibers and special curves on the conic bundle
on the rational surface.
