More on the Skitovich–Darmous theorem for finite Abelian groups

The following theorem is proved. Let $X$ be a finite Abelian group and $\xi_1, \xi_2$ be independent random variables with values in $X$ and with distributions $\mu_1, \mu_2$. Then the independence of the linear statistics $L_1=\alpha_1(\xi_1) + \alpha_2(\xi_2)$ and $L_2=\beta_1(\xi_1) + \beta_2(\xi_2)$, where $\alpha_j, \beta_j$ are automorphisms of the group $X$, implies that $\mu_1,\mu_2$ are idempotent distributions.