Involutory permutation automorphisms of binary linear codes

We investigate the properties of binary linear codes of even length whose permutation automorphism group is a cyclic group generated by an involution. Up to dimension or co-dimension 4, we show that there is no quasi-group code whose permutation automorphism group is isomorphic to C-2. By generalizing the method we use to prove this result, we obtain results on the structure of putative extremal self-dual [16, 36, 72] and [20, 48, 96] codes in the presence of an involutory permutation automorphism.