A note on the automorphism group of Kodaira surfaces

In this note we show that any lift of an automorphism of a primary or secondary Kodaira surface to the universal cover ${\mathbb{C}}^2$ is an affine transformation and, vice versa, we give necessary and sufficient conditions on the coefficients of an affine transformation of the plane to be the lift of an automorphism of some primary Kodaira surface. As a consequence we give a precise description of the structure of the automorphism group of primary and secondary Kodaira surfaces, describing both the connected component of the identity and the group of connected components explicitly.