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

Let $K$ be an $(n-1)$-dimensional piecewise linear sphere on $\left[m\right]$, where $m⩽n+4$. There are a canonical action of $m$-dimensional torus $T^m$ on the moment-angle complex $Z_K$, and a canonical action of $Z_2^m$ on the real moment-angle complex $R{Z}_K$, where $Z_2$ is the additive group with two elements. We prove that any subgroup of $Z_2^m$ acting freely on $R{Z}_K$ is induced by a subtorus of $T^m$ acting freely on $Z_K$. The proof primarily utilizes a suitably modified method of toric wedge induction and the combinatorial structure of the universal complex. As a byproduct, this solves the toric lifting problem for piecewise linear spheres with $m-n⩽4$.