It is shown that an equivariant statistic $S$ is invariantly sufficient iff the generated $\sigma$-algebra and the $\sigma$-algebra of the invariant Borel sets are independent, and that if $S$ is invariantly sufficient and equivariant, then the Pitman estimator for location parameter $\gamma$ is given by $S - E_0(S)$. For independent $X_1, \cdots, X_n$, the existence of an invariantly sufficient equivariant linear statistic is characterized by the normality of $X_1, \cdots, X_n$. Then, the independence of $X_1, \cdots, X_n$ is replaced by a linear framework in which there are established characterizations of the normality of $X = (X_1, \cdots, X_n)$ by properties (invariant sufficiency, admissibility, optimality) of the minimum variance unbiased linear estimator for $\gamma$.
"Invariantly Sufficient Equivariant Statistics and Characterizations of Normality in Translation Classes." Ann. Statist. 11 (1) 330 - 336, March, 1983. https://doi.org/10.1214/aos/1176346084