Relative uniformly positive entropy of induced amenable group actions

Let G be a countably infinite discrete amenable group. It should be noted that a G-system (X, G) naturally induces a G-system (M(X), G), where M(X) denotes the space of Borel probability measures on the compact metric space X endowed with the weak*-topology. A factor map pi : (X, G) -> (Y, G) between two G-systems induces a factor map pi(similar to): (M(X), G) -> (M(Y), G). It turns out that pi(similar to) is open if and only if pi is open. When Y is fully supported, it is shown that pi has relative uniformly positive entropy if and only if pi(similar to) has relative uniformly positive entropy.