Analogue of Brauer's permutation lemma fails over rationals for every non-cyclic finite group

Statement
Let $$G$$ be a finite group that is not a cyclic group (in particular, not a finite cyclic group). Then, there exists two permutation representation $$\varphi_1, \varphi_2:G \to S_n$$ such that $$\varphi_1$$ and $$\varphi_2$$ are equivalent as linear representations over $$\mathbb{Q}$$ (viewed this way by the embedding of $$S_n$$ in $$GL(n,\mathbb{Q})$$ as permutation matrices) but are not equivalent as permutation representations.

In other words, the analogue of opposite of::Brauer's permutation lemma fails to hold for non-cyclic finite groups.

Similar facts

 * Symmetric group of degree six or higher is not weak subset-conjugacy-closed in general linear group over rationals

Opposite facts

 * Brauer's permutation lemma
 * Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group