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The subject of this paper is the study of various families of quartic K3surfaces 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 ofwhich degenerate into the mentioned lines). The second half of this paper isdedicated to finding the Picard group of a general member of each of thesefamilies, and describing it as a lattice. It turns out that for each family thePicard group of a very general surface is generated by the lines and conicslying on said surface.

The Picard Group of Various Families of $(\mathbb{Z}/2\mathbb{Z})^{4}$-invariant Quartic K3 Surfaces