A characterisation of almost simple groups with socle ${}^2\E_6(2)$ or $\M(22)$

Abstract: We show that the sporadic simple group $\M(22)$, the exceptional group of Lie type ${}^2\E_6(2)$ 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.