Small groups of finite Morley rank with a supertight automorphism

Let $G$ be an infinite simple group of finite Morley rank and of Pr"{u}fer $2$-rank $1$ which admits a supertight automorphism $\alpha$ such that the fixed-point subgroup $C_G({\alpha }^n)$ is pseudofinite for all integers $n>0$. We prove that $G\cong {\mathrm{P}\mathrm{G}\mathrm{L}}_2(K)$ for some algebraically closed field $K$ of characteristic $\mathrm{\ne }2$.