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In this paper, we present some results concerning a decomposition of binary self-dual codes having an automorphism of an order which is the product of two odd prime numbers. These results are applied to construct self-dual [72,36,12] codes with an automorphism of order 15. Furthermore, it is proved that the automorphism group of a putative binary extremal self-dual [72,36,16] code contains at most 28 nontrivial types of automorphisms of odd order. A complete list of these possible types of automorphisms is presented.