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

On Kummer-like surfaces attached to singularity and modular forms