Manifolds realized as orbit spaces of non-free $\mathbb Z_2^k$-actions on real moment-angle manifolds

We consider (non-necessarily free) actions of subgroups $H\subset \mathbb Z_2^m$ on the real moment-angle manifold $\mathbb R\mathcal{Z}_P$ corresponding to a simple convex $n$ polytope $P$ with $m$ facets. The criterion when the orbit space $\mathbb R\mathcal{Z}_P/H$ is a topological manifold (perhaps with a boundary) can be extracted from results by M.A. Mikhailova and C. Lange. For any dimension $n$ we construct series of manifolds $\mathbb R\mathcal{Z}_P/H$ homeomorphic to $S^n$ and series of manifolds $M^n=\mathbb R\mathcal{Z}_P/H$ admitting a hyperelliptic involution $\tau\in\mathbb Z_2^m/H$, that is an involution $\tau$ such that $M^n/\langle\tau\rangle$ is homeomorphic to $S^n$. For any simple $3$-polytope $P$ we classify all subgroups $H\subset\mathbb Z_2^m$ such that $\mathbb R\mathcal{Z}_P/H$ is homeomorphic to $S^3$. For any simple $3$-polytope $P$ and any subgroup $H\subset\mathbb Z_2^m$ we classify all hyperelliptic involutions $\tau\in\mathbb Z_2^m/H$ acting on $\mathbb R\mathcal{Z}_P/H$. As a corollary we obtain that a $3$-dimensional small cover has $3$ hyperelliptic involutions in $\mathbb Z_2^3$ if and only if it is a rational homology $3$-sphere and if and only if it corresponds to a triple of Hamiltonian cycles such that each edge of the polytope belongs to exactly two of them.