Isomorphisms between groups of equivariant homeomorphisms of G-manifolds with one orbit type

Given a compact Lie group $G$, a reconstruction theorem for free $G$-manifolds is proved. As a by-product reconstruction results for locally trivial bundles are presented. Next, the main theorem is generalized to $G$-manifolds with one orbit type. These are the first reconstruction results in the category of $G$-spaces, showing also that the reconstruction in this category is very specific and involved.