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 $\mathbb{R}{\mathcal{Z}}_P\mathrm{/}H$ 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 $\mathbb{R}{\mathcal{Z}}_P\mathrm{/}H$ homeomorphic to $S^n$ and series of manifolds $M^n=\mathbb{R}{\mathcal{Z}}_P\mathrm{/}H$ admitting a hyperelliptic involution $\tau \in {\mathbb{Z}}_2^m\mathrm{/}H$, that is an involution $\tau$ such that $M^n\mathrm{/}⟨\tau ⟩$ is homeomorphic to $S^n$. For any simple $3$-polytope $P$ we classify all subgroups $H\subset {\mathbb{Z}}_2^m$ such that $\mathbb{R}{\mathcal{Z}}_P\mathrm{/}H$ 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 $\tau \in {\mathbb{Z}}_2^m\mathrm{/}H$ acting on $\mathbb{R}{\mathcal{Z}}_P\mathrm{/}H$. 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 correspond to a triple of Hamiltonian cycles such that each edge of the polytope belongs to exactly two of them.