Extension properties of orbit spaces for proper actions revisited

Let G be a locally compact Hausdorff group. We study orbit spaces of equivariant absolute neighborhood extensors ( G -ANE's) for the class of all proper G -spaces that are metrizable by a G -invariant metric. We prove that if a proper G -space X is a G -ANE and all G -orbits in X are metrizable, then the G -orbit space X / G is an ANE. Besides, if G is either a Lie group or an almost connected group, then for any closed normal subgroup H of G , the H -orbit space X / H is a G / H -ANE provided that all H -orbits in X are metrizable. If, in addition, H is compact then the hypothesis about the properness of the G -space X can be omitted.