A characterisation of almost simple groups with socle ${}^2{\text{E}}_6(2)$ or $\text{M}(22)$

We show that the sporadic simple group $\text{M}(22)$, the exceptional group of Lie type ${}^2{\text{E}}_6(2)$ and their automorphism groups are uniquely determined by the approximate structure of the centralizer of an element of order 3 together with some information about the fusion of this element in the group.