On element orders in coverings of the simple groups $L_n(q)$ and $U_n(q)$

We prove that if a finite simple linear or unitary group defined over a field of characteristic $p$ and having dimension sufficiently large as compared with $p$ acts on a finite-dimensional vector space over some field of the same characteristic $p$, then the corresponding semidirect product contains an element whose order is distinct from any element order of the simple group.