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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 length . The main theorem proved is that an extremal self-dual doubly even code of length with 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 length and an extremal quaternary code of length arc yielded by techniques used in the proof.