Semi-free actions of zero-dimensional compact groups on Menger compacta

Let $\mu ^{n}$ be the $n$-dimensional universal Menger compactum, $X$ a $Z$-set in $\mu ^{n}$ and $G$ a metrizable zero-dimensional compact group with $e$ the unit. It is proved that there exists a semi-free $G$-action on $\mu ^{n}$ such that $X$ is the fixed point set of every $g \in G \smallsetminus \{e\}$. As a corollary, it follows that each compactum with $\dim \leqslant n$ can be embedded in $\mu ^{n}$ as the fixed point set of some semi-free $G$-action on $\mu ^{n}$.