  The subject of this paper is the study of various families of quartic K3
surfaces which are invariant under a certain $(\mathbb{Z}/2\mathbb{Z})^{4}$
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 (some of
which degenerate into the 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 said surface.
