By a classical theorem due to Mackey, every continuous homomorphism from a locally compact separable group G into Aut(X,µ) (the group of measure preserving transformations) has a point realization, that is, arises from a measure preserving action of G on (X,µ). Recently it was shown by Glasner, Tsirelson and Weiss that also every continuous homomorphism from a subgroup of the group of permutations of the natural numbers admits a point realization. The class of groups of isometries of locally compact separable metric spaces properly contains the above two classes of groups. We show that the result holds here as well. The solution to Hilbert’s fifth problem plays an essential role in our investigations. As a byproduct, we obtain a new characterization of groups of isometries of locally compact separable metric spaces.