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We investigate the properties of binary linear codes of even length whosepermutation automorphism group is a cyclic group generated by an involution. Upto dimension or co-dimension $4$, we show that there is no quasi group codewhose permutation automorphism group is isomorphic to $C_2$. By generalizingthe method we use to prove this result, we obtain results on the structure ofputative extremal self-dual $[72, 36, 16]$ and $[96, 48, 20]$ codes in thepresence of an involutory permutation automorphism.

Involutory permutation automorphisms of binary linear codes