The modular automorphisms of quotient modular curves

We obtain the modular automorphism group of any quotient modular curve of level N, with 4,9 N. In particular, we obtain some non-expected automorphisms of order 3 that appear for the quotient modular curves when the Atkin-Lehner involution w₂₅ belongs to the quotient modular group, such automorphisms are not necessarily defined over Q. As a consequence of the results, we obtain the full automorphism group of the quotient modular curve X₀^*(N²), for sufficiently large N.