Extension of locally pseudocompact group actions

It is shown that: (1) any action of a Moscow group G on a first countable, Dieudonné complete (in particular, on a metrizable) space X can uniquely be extended to an action of the Dieudonné completion γG on X, (2) any action of a locally pseudocompact topological group G on a b f-space (in particular, on a first countable space) X can uniquely be extended to an action of the Weil completion Ḡ on the Dieudonné completion γX of X. As a consequence, we obtain that, for each locally pseudocompact topological group G, every G-space with the b f-property admits an equivariant embedding into a compact Hausdorff G-space. Furthermore, for each pseudocompact group G, every metrizable G-space has a G-invariant metric compatible with its topology. We also give a direct construction of such an invariant metric.