Extension of the pseudo-compact group actions

It is proved that for a given pseudo-compact Hausdorff group $G$, every continuous action $\alpha\colon\,G\times X\to X$ on a metrizable space $X$ has a unique extension to a continuous action $\tilde{\alpha}\colon\,\beta G\times X\to X$, where $\beta G$ is the Stone–Cech extension of $G$.