Manifolds realized as orbit spaces of non-free Z₂ᵏ-actions on real moment-angle manifolds

We consider (non-necessarily free) actions of subgroups H⊂ Z₂ᵐ on the real moment-angle manifold RZP corresponding to a simple convex n polytope P with m facets. The criterion when the orbit space RZP/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 RZP/H homeomorphic to Sⁿ and series of manifolds Mⁿ= RZP/H admitting a hyperelliptic involution τ∈ Z₂ᵐ/H, that is an involution τ such that Mⁿ/τ is homeomorphic to Sⁿ. For any $3$-polytope P we classify all subgroups H⊂ Z₂ᵐ such that RZP/H is a $3$-sphere or a rational homology $3$-sphere. For any $3$-polytope P and any subgroup H⊂ Z₂ᵐ we classify all hyperelliptic involutions τ∈ Z₂ᵐ/H acting on RZP/H. As a corollary we obtain that a $3$-dimensional small cover has $3$ hyperelliptic involutions in Z₂³ 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.