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 $m\le n+4$. There are a canonical action of $m$-dimensional torus $T^m$ on the moment-angle complex ${\mathcal{Z}}_K$, and a canonical action of ${\mathbb{Z}}_2^m$ on the real moment-angle complex $\mathbb{R}{\mathcal{Z}}_K$, where ${\mathbb{Z}}_2$ is the additive group with two elements. We prove that any subgroup of ${\mathbb{Z}}_2^m$ acting freely on $\mathbb{R}{\mathcal{Z}}_K$ is induced by a subtorus of $T^m$ acting freely on ${\mathcal{Z}}_K$. The proof primarily utilizes a suitably modified method of toric wedge induction and the combinatorial structure of a specific binary matroid of rank $4$.