  We study K3 surfaces over a number field $k$ which are double covers of
extremal rational elliptic surfaces. We provide a list of all elliptic
fibrations on certain K3 surfaces together with the degree of a field extension
over which each genus one fibration is defined and admits a section. We show
that the latter depends, in general, on the action of the cover involution on
the fibers of the genus 1 fibration.
