Automorphisms of codes with applications to extremal doubly even codes of length 48

General results on automorphisms of self-dual binary codes are given. These results are applied to the study of extremal self-dual doubly even binary codes of length48. The main theorem proved is that an extremal self-dual doubly even code of length48with a nontrivial automorphism of odd order is equivalent to the extended quadratic residue code. Interesting constructions of the binary extended Golay code as well as a conjecture about a possible connection between an extremal self-dual doubly even code of length72and an extremal quaternary code of length24arc yielded by techniques used in the proof.