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.

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: $$(gh)^n=ag^n a^{-1}bh^n b^{-1}$$