The modular automorphisms of quotient modular curves

We obtain the modular automorphism group of any quotient modular curve of level $N$, with $4,9\nmid N$. In particular, we obtain some non-expected automorphisms of order 3 that appear for the quotient modular curves when the Atkin-Lehner involution $w_{25}$ belongs to the quotient modular group, such automorphisms are not necessarily defined over $\mathbb{Q}$. As a consequence of the results, we obtain the full automorphism group of the quotient modular curve $X_0^{\ast }(N^2)$, for sufficiently large $N$.