Galois points on quartic surfaces

Let S be a smooth hypersurface in the projective three space and consider a projection of S from P is an element of S to a plane H. This projection induces an extension of fields k(S)/k(H). The point P is called a Galois point if the extension is Galois. We study structures of quartic surfaces focusing on Galois points. We will show that the number of the Galois points is zero, one, two, four or eight and the existence of some rule of distribution of the Galois points.