Curves with more than one inner Galois point

Let C be an irreducible plane curve of PG(2,K) where K is an algebraically closed field of characteristic p≥0. A point Q∈C is an inner Galois point for C if the projection πQ from Q is Galois. Assume that C has two different inner Galois points Q1 and Q2, both simple. Let G1 and G2 be the respective Galois groups. Under the assumption that Gi fixes Qi, for i=1,2, we provide a complete classification of G=〈G1,G2〉 and we exhibit a curve for each such G. Our proof relies on deeper results from group theory.