Isometry groups of Polish ultrametric spaces

We solve a long-standing open problem, formulated by Krasner in the 1950's, in the context of Polish (i.e. separable complete) ultrametric spaces by providing a characterization of their isometry groups using suitable forms of generalized wreath products of full permutation groups. Since our solution is developed in the finer context of topological (Polish) groups, it also solves a problem of Gao and Kechris from 2003. Furthermore, we provide an exact correspondence between the isometry groups of Polish ultrametric spaces belonging to some natural subclasses and various kinds of generalized wreath products proposed in the literature by Hall, Holland, and Malicki.