Exceptional action of the simple groups $L_4(q)$ in the defining characteristic

We find a counterexample to Problem 14.60 from the Kourovka notebook by giving an example of a group $\rm {PSL}_4(q)$ which is not recognizable among its covers from spectrum. Namely, we show that such a group has a module in the defining characteristic with the property that the element order set of the corresponding semidirect product equals that of the group itself.