= Solution
The <valency theorem> states that for a nonconstant analytic map between compact connected Riemann surfaces,
$$
\sum_{z\in f^{-1}(w)}m_f(z)=\deg f
$$
for every target point $w$.
For a rational map $f=p/q$ on the Riemann sphere, first cancel common factors; then the <degree of a rational map of the Riemann sphere> is
$$
\deg f=\max(\deg p,\deg q).
$$
The analytic isomorphisms are exactly the degree-one maps, namely the <Möbius transformations>.
The required transformation is
$$
\boxed{h(z)=\frac{z+1}{z-1}}.
$$
It sends $\infty\leftrightarrow1$ and $0\leftrightarrow-1$. Its fixed points satisfy
$$
\boxed{z^2-2z-1=0}.
$$
The <octahedral rotation orbits on the Riemann sphere> have possible sizes
$$
\boxed{6,\ 8,\ 12,\ 24},
$$
corresponding respectively to vertices, face centres, edge centres, and generic points.
In the displayed $F$, no numerator factor vanishes at $0$ or at a fourth root of one. Hence $0,\pm1,\pm i$ are poles of order four. The numerator and denominator have degrees $24$ and $20$, so $F(z)\sim z^4$ at infinity; infinity is also a pole of order four. Thus
$$
F^{-1}(\infty)=\{0,\infty,\pm1,\pm i\},
\qquad m_F=4\text{ at each point},
$$
and
$$
\boxed{\deg F=24}.
$$
Finally let $z$ have stabilizer of order $e$. A local coordinate turns its stabilizer action into rotation by $e$th roots of unity. Since $F$ is invariant, the first nonconstant term of its local expansion has exponent divisible by $e$, so $m_F(z)\geq e$. The orbit has $24/e$ points and therefore contributes at least $24$ to the fibre through $F(z)$. The valency theorem and $\deg F=24$ leave no room for any further point. The <degree-sized invariant separates finite-group orbits>, so $F(z)=F(w)$ implies that $z$ and $w$ lie in the same orbit.
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