Let $T=\operatorname{PSL}_n(q)$ with $n$ a prime number. Then the $\mathscr{C}_3$ subgroup $M=\langle x\rangle{:}\langle\sigma\rangle$ of $T$ is isomorphic to $\mathbb{Z}_{\frac{q^n-1}{(q-1)(n,q-1)}}{:}\mathbb{Z}_n$, where $x$ comes from the Singer cycle.

Note that $\sigma$ has a matrix which is a permutation matrix corresponding to $(1,2,\dots,n)$ in $\operatorname{SL}_n(q)$. It follows that $\langle\sigma\rangle$ is preserved by any outer automorphism of $T$.

My question is: is $\langle x\rangle$ preserved by any outer automorphism of $T$? It is true for the diagonal automorphism and the $\mathscr{C}_3$ subgroup of $\operatorname{PGL}_n(q)$ is isomorphic to $\mathbb{Z}_{\frac{q^n-1}{q-1}}{:}\mathbb{Z}_n$. How about the field automorphism and the graph automorphism? How can we find a matrix form of $x$ in this case?

That is to say, if $o\le\operatorname{Out}(T)$, then is the $\mathscr{C}_3$ subgroup of $T.o$ just $M.o$? And how about the case when $T=\operatorname{PSU}_n(q)$ and $M$ is isomorphic to $\mathbb{Z}_{\frac{q^n+1}{(q+1)(n,q+1)}}{:}\mathbb{Z}_n$, where $n$ is still a prime number? In this case we only need to consider the field automorphism.