On the Heyde theorem for finite Abelian groups

It is well-known Heyde's characterization theorem for the Gaussian distribution on the real line: if xij are independent random variables, alphaj, betaj are nonzero constants such that betaialphai(-1) +/- betajalphaj(-1) not equal 0 for all i not equal j and the conditional distribution of L-2 = beta(1)xi(1) + ... + beta(n)xi(n) given L-1 = alpha(1)xi(1) + ... + alpha(n)xi(n) is symmetric, then all random variables j are Gaussian. We prove some analogs of this theorem, assuming that independent random variables take on values in a finite Abelian group X and the coefficients alphaj, betaj are automorphisms of X.