Properties of element orders in covers for $\operatorname L_n(q)$ and $\operatorname U_n(q)$

We show that if a finite simple group $G$, isomorphic to $\operatorname{PSL}_n(q)$ or $\operatorname{PSU}_n(q)$ where either $n\ne4$ or $q$ is prime or even, acts on a vector space over a field of the defining characteristic of $G$; then the corresponding semidirect product contains an element whose order is distinct from every element order of $G$. We infer that the group $\operatorname{PSL}_n(q)$, $n\ne4$ or $q$ prime or even, is recognizable by spectrum from its covers thus giving a partial positive answer to Problem 14.60 from the Kourovka Notebook.