Nonexistence of binary quasi group codes with involutory permutation automorphism group

Let $C$ be a binary quasi group code of length $n=2m>2$, and let $\operatorname{PAut}(C)$ denote its permutation automorphism group. Nonexistence conjecture. There is no binary quasi group code $C$ of length $n=2m>2$ such that

$$\operatorname{PAut}(C)\cong C_2.$$

This conjecture generalizes the preceding nonexistence result for lengths $2<n=2m\leq 8$; the authors report that computer calculations suggest it, but a proof would require handling many cases.