A note about dual representations of group actions on Lipschitz-free spaces

Let F (M) be the Lipschitz-free space of a pointed metric space M. For every isometric continuous group action of G we have an induced continuous dual action on the weak-star compact unit ball B₅ (₌) ^* of the dual space Lip₀ (M) =F (M) ^*. We pose the question when a given abstract continuous action of G on a topological space X can be represented through a G-subspace of B₅ (₌) ^*. One of such natural examples is the so-called metric compactification (of isometric G-spaces) for a pointed metric space. As well as the Gromov G-compactification of a bounded metric G-space. Note that there are sufficiently many representations of compact G-spaces on Lipschitz-free spaces.