The equivariant universality and couniversality of the Cantor cube

Let (G, X, α) be a G-space, where G is a non-Archimedean (having a local base at the identity consisting of open subgroups) and second countable topological group, and X is a zero-dimensional compact metrizable space. Let (H ({0,1} א₀), {0,1} א₀,T) be the natural (evaluation) action of the full group of autohomeomorphisms of the Cantor cube. Then (1) there exists a topological group embedding φ : G (Rightwards arrow with hook) H({0,1} א₀); (2) there exists an embedding ψ: X ¬ {0,1} א₀, equivariant with respect to φ, such that ψ (X) is an equivariant retract of {0,1}א₀ with respect to φ and ψ.