Exceptional points on Atkin--Lehner quotients

We study the rational points on the star curve $X_0^*(N) := X_0(N)/W(N)$, the quotient of the classical modular curve $X_0(N)$ by the full group of Atkin--Lehner involutions, for squarefree levels $N$. Rational points on $X_0^*(N)$ parameterize $\mathbb{Q}$-curves, i.e.\ elliptic curves $E/\overline