  We study a family of lattice polarized $K3$ surfaces which is an extension of
the family of Kummer surfaces derived from principally polarized Abelian
surfaces. Our family has two special properties. First, it is coming from a
resolution of a simple $K3$ singularity. Second, it has a natural
parametrization by Hermitian modular forms of four complex variables. In this
paper, we show two results: (1) We determine the transcendental lattice and the
N\'eron-Severi lattice of a generic member of our family. (2) We give a
detailed description of the double covering structure associated with our $K3$
surfaces.
