Are representation isomorphisms of permutation representations necessarily permutation matrices?

Question

Let $G$ be a group and $\pi:G\rightarrow GL_n(\mathbb{C})$ be its permutation representation. Let $U$ be a unitary matrix commuting with $\pi$. That is, $U$ is an automorphism of the $\mathbb{C}G$ module. The question is: is the action of $U$ necessarily a permutation on $\mathbb{C}G$, maybe up to a complex factor?

Thanks.