How to solve for a homeomorphism $h : [-1, 1] \to [0, 1]$ that conjugates the mappings $P_2 : [-1, 1] \to [-1, 1],$ $x \mapsto 2x^2 - 1$ and $F_4 : [0, 1] \to [0, 1],$ $x \mapsto 4x(1-x)$? That is $$F_4 \circ h = h \circ P_2.$$
I tried using the fact that $P_2(\cos \theta) = \cos 2\theta$ but couldn't evaluate anything useful.