Solution of a functional equation on compact groups using Fourier analysis

Let $G$ be a compact group, let $n\in N\setminus \{0,1\}$ be a fixed element and let $\sigma$ be a continuous automorphism on $G$ such that ${\sigma }^n=I$. Using the non-abelian Fourier transform, we determine the non-zero continuous solutions $f:G\to C$ of the functional equation $$f(xy)+\sum _{k=1}^{n-1}f({\sigma }^k(y)x)=nf(x)f(y),x,y\in G,$$ in terms of unitary characters of $G$.