On recognizability of $\operatorname{PSU}_3(q)$ by the orders of maximal abelian subgroups

Li and Chen in 2012 proved that the simple group $A_1(p^n)$ is uniquely determined by the set of orders of its maximal abelian subgroups. Later the authors proved that if $L=A_2(q)$, where $q$ is not a Mersenne prime, then every finite group with the same orders of maximal abelian subgroups as $L$ is isomorphic to $L$ or an extension of $L$ by a subgroup of the outer automorphism group of $L$. In this paper, we prove that if $L=\operatorname{PSU}_3(q)$, where $q$ is not a Fermat prime, then every finite group with the same set of orders of maximal abelian subgroups as $L$ is an almost simple group with socle $\operatorname{PSU}_3(q)$.