The Picard group of various families of (Z/2Z)^4-invariant quartic K3 surfaces

The subject of this paper is the study of various families of quartic K3 surfaces which are invariant under a certain (Z/2Z)^4 action. In particular, we describe families whose general member contains 8,16,24 or 32 lines as well as the 320 conics found by Eklund (2010) (some of which degenerate to the above mentioned lines). The second half of this paper is dedicated to finding the Picard group of a general member of each of these families, and describing it as a lattice. It turns out that for each family the Picard group of a very general surface is generated by the lines and conics lying on the surface.