Brauer's permutation lemma states that any two permutation matrices are conjugate in $\mathrm{GL}(n,\mathbb{C})$ if and only if they are conjugate in the symmetric group, i.e., they have the same cycle type (we can replace $\mathbb{C}$ by any field of characteristic zero).

Another way of stating it is that any two permutation representations of a finite cyclic group that are conjugate as linear representations in characteristic zero are in fact equivalent as permutation representations.

My question: for what classes of finite groups does this result hold? i.e., for what classes of finite groups is it true that any two permutation representations that are conjugateas linear representations in characteristic zero are in fact equivalent as permutation representations?

I think some counterexamples involving Mathieu groups exist, so this is not true for all finite groups. What I was looking for are theorems of the form that it is true for all groups satisfying some well-studied property or in some well-identified class.

One application of this would be as follows: suppose $G$ is a finite group such that the quotient of the automorphism group of $G$ by the group of class-preserving automorphisms of $G$ satisfies the above condition. (Class-preserving automorphisms are automorphisms that preserve each conjugacy class; for a finite group, this is equivalent to preserving all the characters). Then, the orbit sizes under the action of the automorphism group of $G$ are the same for the set of conjugacy classes and the set of (equivalence classes of) irreducible representations.