Suppose you have $G$ a finite group of Lie type (say $\mathrm{Sp}_4( \mathbb{F}_5)$ as a case I particularly care about, but there are others.) Your friend picks a subgroup $H$ and selects random matrices from it, takes their characteristic polynomials, and hands them to you. Is there a criterion that will let you figure out if $H=G$?

Existing theorems I've found use the existence of transvections, which can't (as far as I can tell) be deduced from the above. Or they work over a field of characteristic $0$, where we have more Galois groups to work with. Some also work over the algebraic closure, which is a weaker hypothesis and I'm hoping it strengthens this result. This has applications to large image theorems and experimental mathematics.