The $(2)^4$ Gepner model is an example of a superconformal field theory (SCFT) on a smooth quartic $K3$ surface. This model is very well understood, while in general it remains an open problem how to construct SCFTs on smooth quartic $K3$ surfaces. We present a construction of a smooth four parameter family of SCFTs on a family of smooth quartic $K3$ surfaces. Although we do not obtain new SCFTs per se, these theories are important because they yield deformations of the Gepner model $(2)^4$ which are thus accessible to techniques that are normally restricted to this latter model.