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We complete the computation of all $\mathbb{Q}$-rational points on all the$64$ maximal Atkin-Lehner quotients $X_0(N)^*$ such that the quotient ishyperelliptic. To achieve this, we use a combination of various methods, namelythe classical Chabauty--Coleman, elliptic curve Chabauty, quadratic Chabauty,and the bielliptic quadratic Chabauty method combined with the Mordell-Weilsieve. Additionally, for square-free levels $N$, we classify all$\mathbb{Q}$-rational points as cusps, CM points (including their CM field and$j$-invariants) and exceptional ones. We further indicate how to use this tocompute the $\mathbb{Q}$-rational points on all of their modular coverings.

Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings