Assume $p=(q^n-1)/(q-1)$ with $p$ prime and let $V:=\mathbb F^n_q$ denote an $n$-dimensional vector space over $\mathbb F_q$. There is a natural action of $PSL(n,q)$ on the $(q^n-1)/(q-1)$ one-dimensional subspaces of $V$ as each element in $PSL(n,q)$ is an invertible $n \times n$ matrix corresponding to a linear tranformation $V \rightarrow V$. Such action allows us to associate $PSL(n,q)$ with a transitive permutation group of prime degree $p$.
I read that almost simple groups $G$ satisfying $PSL(n,q)\leq G\leq {\rm Aut}(PSL(n,q))$ also act transitively and faithfully on $p=(q^n-1)/(q-1)$ one-dimensional subspaces of $V:=\mathbb F^n_q$. That leads to more transitive permutation groups of degree $p$ (which is the main goal of my study). However, I fail to understand how to construct such action. Any help would be appreciated.
I tried, though unsuccessfully to get some understanding here:
Point stabilisers in almost simple groups
http://www.math.rwth-aachen.de/~Gerhard.Hiss/Students/MasterarbeitFengler.pdf
Pages 39-40 provide the construction of such action, but I have problems in understanding why it is faithful.