On Extremal Self-Dual Codes of Length 96

Let C be a binary extremal self-dual code of length 96. We prove that an automorphism of C of order 3 has 6 or no fixed points and an automorphism of order 5 has 6 fixed points. Moreover, if all automorphisms of order 3 are fixed point free then Aut(C) is solvable and its order divides 2⁵3 or 2⁵5 or Aut(C) is the alternating group A₅ which is the only possible group of order 60. Furthermore, |Aut(C)| = 20 or 40 cannot occur.