We consider the problem of classifying toroidal circle planes with respect to the dimension of their automorphism groups. With tools from topology, we prove that these groups are Lie groups of dimension at most 6. From the results on flat Minkowski planes by Schenkel, we classify planes whose automorphism group has dimension at least 4.

In the case of dimension 3, we propose a framework for the full classification based on all possible geometric invariants of the automorphism group. When the group fixes exactly one point, we characterise two cases completely with a new family of planes called (modified) strongly hyperbolic planes and the family constructed by Artzy and Groh. Using these results, we determine the automorphism group of the planes constructed by Polster.