  We describe two constructions of elliptic K3 surfaces starting from the
Kummer surface of the Jacobian of a genus 2 curve. These parallel the
base-change constructions of Kuwata for the Kummer surface of a product of two
elliptic curves. One of these also involves the analogue of an Inose fibration.
We use these methods to provide explicit examples of elliptic K3 surfaces over
the rationals of geometric Mordell-Weil rank 15.
