Rational representations and permutation representations of finite groups

We investigate the question which ℚ-valued characters and characters of ℚ -representations of finite groups are ℤ-linear combinations of permutation characters. This question is known to reduce to that for quasi-elementary groups, and we give a solution in that case. As one of the applications, we exhibit a family of simple groups with rational representations whose smallest multiple that is a permutation representation can be arbitrarily large