Properties of element orders in covers for Ln(q) and Un(q)

We show that if a finite simple group G , isomorphic to PSL n ( q ) or PSU n ( q ) where either n ≠ 4 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 PSL n ( q ), n ≠ 4 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.