Equivariant extension properties of coset spaces of locally compact groups and approximate slices

We prove that for a compact subgroup H of a locally compact Hausdorff group G, the following properties are mutually equivalent: (1) G/H is finite-dimensional and locally connected, (2) G/H is a smooth manifold, (3) G/H satisfies the following equivariant extension property: for every paracompact proper G-space X having a paracompact orbit space, every G-map A→G/H from a closed invariant subset A⊂X extends to a G-map U→G/H over an invariant neighborhood U of A. A new version of the Approximate Slice Theorem is also proven in the light of these results.