On the automorphism groups of binary linear codes

Let C be a binary linear code and suppose that its automorphism group contains a non trivial subgroup G. What can we say about C knowing G? In this paper we collect some answers to this question in the cases G=Cp, G=C2p and G=D_2p (p an odd prime), with a particular regard to the case in which C is self-dual. Furthermore we generalize some methods used in other papers on this subject. Finally we give a short survey on the problem of determining the automorphism group of a putative self-dual [72,36,16] code, in order to show where these methods can be applied.