An isogeny of K3 surfaces

In a recent paper Ahlgren, Ono and Penniston described the L-series of K3 surfaces from a certain one parameter family in terms of those of a particular family of elliptic curves. The Tate conjecture predicts the existence of a correspondence between these K3 surfaces and certain Kummer surfaces related to these elliptic curves. A geometric construction of this correspondence is given here, using results of D. Morrison on Nikulin involutions.