Permutation-like matrix groups

Let G be a group of complex n × n matrices. We call G a permutation-like group if every matrix in G is similar to a permutation matrix. We consider the question whether a permutation-like group is equivalent to a group of permutation matrices. Examples show that this is not always the case. We investigate this problem in detail for some small values of n and conjecture that under some additional condition the above implication holds.