  Let $G$ be an infinite simple group of finite Morley rank and $\alpha$ a
supertight automorphism of $G$ so that the fixed point subgroup
$P_n:=C_G(\alpha^n)$ is pseudofinite for all $n\in \mathbb{N}\setminus\{0\}$.
It is know (using CFSG) that the socle $S_n:={\rm Soc}(P_n)$ is a (twisted)
Chevalley group over a pseudofinite field. We prove that there is $r\in
\mathbb{N}\setminus\{0\}$ so that for each $n$ we have $[P_n:S_n] < r$ and that
there is no $m \in \mathbb{N}\setminus \{0\}$ so that for each $n$ the sizes of
the Sylow $2$-subgroups of $S_n$ are bounded by $m$. We also note that in the
recent identification result of $G$ under the assumption ${\rm pr}_2(G)=1$, the
use of CFSG is not needed.
