Union of equivariant extensors and equivariant covering spaces

Let G be a compact topological group and P be the class of all paracompact and Hausdorff G-spaces. We prove that a paracompact space that is a local G-AN E(P) is a G-AN E(P). Then we give an application to the theory of equivariant overlays by proving the following: Let p : [...] --> X be G-overlays of a metrizable connected G-space X. Then an equivariant map f : Z --> X from Z, a metrizable connected G-space, can be lifted to an equivariant map F : Z --> [...] if and only if f[...].