  We construct geometric isogenies between three types of two-parameter
families of K3 surfaces of Picard rank 18. One is the family of Kummer surfaces
associated with Jacobians of genus-two curves admitting an elliptic involution,
another is the family of Kummer surfaces associated with the product of two
non-isogeneous elliptic curves, and the third is the twisted Legendre pencil.
The isogenies imply the existence of algebraic correspondences between these K3
surfaces and prove that the associated four-dimensional Galois representations
are isomorphic. We also apply our result to several subfamilies of Picard rank
19. The result generalizes work of van Geemen and Top.
