Permutation-like matrix groups with a maximal cycle of length power of two

If every element of a matrix group is similar to a permutation matrix, then it is called a permutation-like matrix group. References [3], [4] and [5] showed that, if a permutation-like matrix group contains a maximal cycle such that the maximal cycle generates a normal subgroup and the length of the maximal cycle equals a prime, or a square of a prime, or a power of an odd prime, then the permutation-like matrix group is similar to a permutation matrix group. In this paper, we prove that if a permutation-like matrix group contains a maximal cycle such that the maximal cycle generates a normal subgroup and the length of the maximal cycle equals any power of 2, then it is similar to a permutation matrix group.