On element orders in covers of $L_4(q)$ and $U_4(q)$

Suppose that $L$ is one of the finite simple groups $\operatorname{PSL}_4(q)$ or $\operatorname{PSU}_4(q)$ and $L$ acts on a vector space $W$ over a field whose characteristic divides $q$. We prove that the natural semidirect product of $W$ and $L$ contains an element whose order differs from the order of any element of $L$, thus answering questions 14.60 and 17.73 (a) of the Kourovka Notebook.