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 a manifold, (2) \(G/H\) is finite-dimensional and locally connected, (3) \(G/H\) is locally contractible, (4) \(G/H\) is an ANE for paracompact spaces, (5) \(G/H\) is a metrizable \(G\)-ANE for paracompact proper \(G\)-spaces having a paracompact orbit space. A new version of the Approximate slice theorem is also proven in the light of these results.