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