If a finite group acts faithfully and primitively on the points of a configuration, will the figure be flag transitive?

Question

Suppose we have a finite group G, and we had a configuration of points, lines etc. and lets say that the group acts faithfully and primitively on the set of points. Will the group be guaranteed to be flag-transitive on the entire figure or can G fix some proper subset of lines, for instance? My intuition says that shouldn't be possible, but I can't figure out how to prove it.

Answer 1

Define a graph as follows. The points are the vertices of a $p$-sided polygon for some prime $p \ge 7$, and two points are joined by an edge if their distance apart in the polyagon is at most $2$. So each point is joined by an edge to exactly four other points.

It is not hard to see that the automorphism group of this graph is the dihedral group of order $2p$. It acts transitively on points and hence also primitively, because $p$ is prime, but it has two orbits on lines, namely the lines joining adjacent points in the polygon, and the lines joining points at distance $2$ apart.