Real Kummer quartics and their Heisenberg invariance

We consider two classifications of real Kummer quartics. They use the Heisenberg invariance of Kummer quartics. The first divides the whole variety of real Kummer quartics into four classes according to the Heisenberg-invariance type and then subdivides each class into subclasses to obtain a deformation classification. This subdivision into subclasses is performed by means of the topological classification of the real parts of real Kummer quartics. The second classification deals with the set of real Kummer quartics with a fixed Heisenberg group. Such a set consists of a continuous part and a discrete part. We describe the deformation classes of the continuous part and describe its discrete part.