Let $G=PSL(n,q)$ be the projective linear group over $\mathbb{F}_q$ and let $\sigma$ be an **outer** automorphism of $G$. (The description of outer automorphism group of $PSL(n,q)$ is well-known, see for example Wilson's
book, Theorem 3.2, page 50.)

The group $G$ acts on $G$ by $g\cdot h=gh\sigma(g^{-1})$ for all $g,h\in G$.
The orbits of this action are called *$\sigma$-twisted conjugacy classes*.

My question is the following:

Is it possible to compute (in terms of the sizes of the classes of $G$) the sizes of these twisted conjugacy classes?

Possible idea:

Let $g\in G$. The $\sigma$-twisted conjugacy class of $g$ can be realized as a conjugacy class of the semidirect product $G\rtimes\langle\sigma\rangle$.

In this context, the question would be the following:

Is it possible to compute the sizes of the conjugacy classes of the group $G\rtimes\langle\sigma\rangle$?

The motivation for my question is the following:

Let $p$ be a prime number. Since there are no conjugacy classes of size $2p$ in finite simple groups (see this post), I would like to prove that there are no twisted conjugacy classes of $PSL(n,q)$ of size $2p$.