Is $\mathbf{PGL}_2(k)$ a characteristic subgroup of $\mathbf{P\Gamma L}_2(k)$, $k$ a division ring?

Question

Let $k$ be a division ring, and consider the projective semilinear group $\mathbf{P\Gamma L}_2(k) =: L$.

We know that $L$ has a naturally defined normal subgroup isomorphic to $\mathbf{PGL}_2(k)$ (if we introduce $\mathbf{P\Gamma L}_2(k)$ as $\mathbf{PGL}_2(k) \rtimes \mathrm{Aut}(k)$).

Is it also a characteristic subgroup of $L$ ?

(If I am not mistaken, this is the case if $k$ is a finite field, possibly up to some small counter examples.)