What are the necessary conditions for a polynomial Q(X) such that the roots of Q(X) - X are equal to the real roots of a polynomial P?

Question

If $P(X), Q(X) ∈ ℝ[X]$ , and $P(X) | P( Q(X) ) $ , what could be the necessary conditions for $Q(X)$ such that the set of the real roots of $P(X) $ to be equal to the set of the real roots of $Q(X) - X $( i.e. the set of fixed points of the polynomial function of $Q(X)$ ) ?

Answer 1

One necessary condition is that $Q$ does not induce a permutation without fixed points on any finite subset of $\mathbb R$, i.e. there does not exist a finite set $S \subset \mathbb R$ such that $Q(S) = S$ but $Q(s) \ne s$ for all $s \in S$. Namely, if such $S$ existed we could take $P(X) = \prod_{s \in S} (X - s)$.