Curves with more than one inner Galois point

Let $\mathcal{C}$ be an irreducible plane curve of $\text{PG}(2,\mathbb{K})$ where $\mathbb{K}$ is an algebraically closed field of characteristic $p\geq 0$. A point $Q\in \mathcal{C}$ is an inner Galois point for $\mathcal{C}$ if the projection $\pi_Q$ from $Q$ is Galois. Assume that $\mathcal{C}$ has two different inner Galois points $Q_1$ and $Q_2$, both simple. Let $G_1$ and $G_2$ be the respective Galois groups. Under the assumption that $G_i$ fixes $Q_i$, for $i=1,2$, we provide a complete classification of $G=\langle G_1,G_2 \rangle$ and we exhibit a curve for each such $G$. Our proof relies on deeper results from group theory.