Maximal subgroups of symmetric groups defined on projective spaces over finite fields

Let $P\Gamma L(n, q)$ be a complete projective group of semilinear transformations of the projective space $P(n--1,q)$ of projective degree $n--l$ over a finite field of $q$ elements; we consider the group in its natural 2-transitive representation as a subgroup of the symmetric group $S(P^*(n—1,q))$ on the set $P^*(n-1,q)=P(n-1,q)\setminus\{\overline0\}$. In the present note we show that for arbitrary $n$ satisfying the inequality $n>4\frac{q^n-1}{q^{n-1}-1}$ [in particular, for $n>4(q+1)$] and for an arbitrary substitution $g\in S(P^*(n-1,q))\setminus P\Gamma L(n,q)$ the group $\langle P\Gamma L(n,q),g\rangle$ contains the alternating group $A(P^*(n-1,q))$.
For $q=2,3$ this result is extended to all $n\ge3$.