G-fibrant extensions and twisted products

Let G be a compact metrizable group and H its closed subgroup. We prove that every compact metrizable G  -space over G/HG/H admits a G  -fibrant extension over G/HG/H. This implies that the functor of twisted product G×H−G×H− takes H-fibrant extensions to G  -fibrant extensions. In particular, the twisted product G×HSG×HS is a G-fibrant space when S is a compact H-fibrant space.