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The structure of binary self-dual codes invariant under the action of acyclic group of order $pq$ for odd primes $p\neq q$ is considered. As anapplication we prove the nonexistence of an extremal self-dual $[96, 48, 20]$code with an automorphism of order $15$ which closes a gap in `"On extremalself-dual codes of length 96", IEEE Trans. Inf. Theory, vol. 57, pp. 6820-6823,2011'.

On the Automorphisms of Order 15 for a Binary Self-Dual [96, 48, 20] Code