Invariant estimates of the shift for distributions on finite groups

Let $\{\mu\alpha\}$ be a family of measures on a finite group $G$ where $\mu$ is a measure on $G$ and $\alpha\in G$. The measure $\mu\alpha$ is said to be obtained from $\mu$ by shift $\alpha$.
In the paper, invariant estimates $f(x)$, $x$ being a sample $(x_1,\dots,x_n)$ from the distribution $\mu\alpha$ with a known $\mu$, for the shift $\alpha$ are constructed, i.e. such estimates that $\varphi(f(x))=f(\varphi(x))$ for any automorphism $\varphi$.