In 1976, Hiroshi Inose uncovered a beautiful relation between certain quartic K3 surfaces, including the Fermat quartic, and Kummer surfaces obtained from a product of isogeneous elliptic curves. I review this result and explain how, by applying appropriate dualities, it can be used to explicitly construct a family of associated conformal field theories. In particular this means that for an entire family of K3 surfaces, which is given in terms of algebraic equations, the associated conformal field theories are well understood.