Let $F$ map points in $\mathbb{R}^3$ to points on the unit disk, $\Delta$, in the $xz$-plane (identified with $\mathbb{C}$) by projecting through $\Delta$ along lines that intersect at $(0,-1,0)$. Let $H$ be the forward sheet of the hyperboloid as in the hyperboloid model.

Given an element $p:z\to(az+b)/(cz+d)$ of $\text{PSL}(2,\mathbb{R})$ is there a transformation $g$ of $H$ such that $F \circ g \circ F^{-1} = p$ on $\Delta$?

If the answer to (1) is "yes," then is there a known explicit formula for $g$ in terms of $a$, $b$, $c$, and $d$?

To clarify the situation, in the attached figure $g$ is a transformation of $H$ that nearly satisfies the condition from (1) on the green test curve. I found $g$ in the figure by manually tweaking the elements of a 4x4 matrix until the red (projected) and green (correct) curves became visually close in the disk on the bottom right.