Toric wedge induction and toric lifting property for piecewise linear spheres with a few vertices

Let K be an (n-1) -dimensional piecewise linear sphere on m, where m n+4. There are a canonical action of m-dimensional torus Tᵐ on the moment-angle complex ZK, and a canonical action of Z₂ᵐ on the real moment-angle complex RZK, where Z₂ is the additive group with two elements. We prove that any subgroup of Z₂ᵐ acting freely on RZK is induced by a subtorus of Tᵐ acting freely on ZK. The proof primarily utilizes a suitably modified method of toric wedge induction and the combinatorial structure of a specific binary matroid of rank 4.