Let $G$ be a finite group, and $n$ an integer coprime to $|G|$. Then we have the following map, which is clearly not a morphism of groups in general: $$g\mapsto g^n.$$

This induces a linear automorphism of $\mathbb{Z}[G]^G$, the algebra of $G$-invariant functions on $G$ under convolution, and surprisingly, this induced map is also an algebra automorphism, as can be seen by passing to $\mathbb{C}$ and noting that this is the Galois action on characters, which permutes the set of primitive idempotents of this algebra.

My question is whether this surprising fact can be explained directly, without using character theory? I would be interested in both a high-concept explanation of this symmetry, or a generators and relations argument for why there exists, for any $g,h$ in $G$, an $a,b\in G$ with: $$(gh)^n=ag^n a^{-1}bh^n b^{-1}.$$