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We show that $X$ admits a locally finite $\\sigma$-discrete $G$-functionally open cover each member of which is $G$-homeomorphic to a twisted product $G\\times_H S_i$, where $H$ is a compact large subgroup of $G$ (i.e., the quotient $G/H$ is a manifold). If, in addition, the space of connected components of $G$ is compact and $X$ is normal, then $X$ itself is $G$-homeomorphic to a twisted product $G\\times_KS$, where $K$ is a maximal compact subgroup of $G$. 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