Fundamental Theorem of Projective Geometry for Finite Fields

Question

I was trying to understand the proof that the automorphism group of a projective plane over a finite field $F_q$ (denoted $PG(2, q)$) is $P\Gamma L(3, q)$, ie. elements $PGL(3, q)$ composed with field automorphisms of $F_q$ acting coordinate-wise. It is clear that $P\Gamma L(3, q) \leqslant Aut(PG(2, q))$, and I read that the reverse inclusion is true due to the Fundamental Theorem of Projective Geometry. I have seen a proof of the Fundamental Theorem of Projective Geometry in Artin's "Geometric Algebra," but it uses some deeper concepts in Algebraic Geometry. I was wondering if there is a simpler proof for the finite field case?