We know that the full automorphism group of the $\pi_q = PG(2,q)$ acts imprimitively on the flags (all flags through a fixed point form a block). But, things change when we consider the action of full correlation group (semi-direct product of the automorphism group with cyclic group of order 2). I have been able to show that this group, which I'll call $Cor(\pi_q)$, acts primitively on the flags for $q = 2,3,4,5,7$ using GAP. Is it true for all prime powers $q$?
My question is "For what values of $q$ is the group $Cor(\pi_q)$ flag-primitive?"
For those familiar with the notion of generalized hexagons: original motivation of this was to study some properties known as valuations of the generalized hexagon of order $(q,1)$ constructed from $PG(2,q)$. For that, it's convenient if the automorphism group of hexagon (which is same as the correlation group of the projective plane) acts primitively and distance transitively on its points. For small values, it turned out to be so but I couldn't find a theoretical argument to prove it for some general class of values of $q$.