A compact bordered Klein surface of (algebraic) genus g ≦ 2 is said to have maximal symmetry  if its automorphism group is of order 12(g – 1), the largest possible. An M*-group acts as the automorphism group of a bordered surface with maximal symmetry. M*-groups were first studied in , and additional results about these groups are in [5, 7, 8].
Here we construct a new, interesting family of M*-groups. Each group G in the family is an extension of a cyclic group by the automorphism group of a torus T with holes that has maximal symmetry. Furthermore, G acts on a bordered Klein surface X that is a fully wound covering  of T, that is, an especially nice covering in which X has the same number of boundary components as T. The construction we use for the new family of M*-groups is a standard one that employs group automorphisms to define extensions of groups.