There is some textbook literature which essentially covers the issues raised here, though it often deals with more general situations. Over finite fields life is simpler, since Schur indices are 1. See for example Curtis & Reiner (1962), Section 70, and also the book on character theory by Isaacs (reprinted by AMS).
The basic criterion deals with an arbitrary finite group $G$ and its representations over a finite splitting field $\mathbb{F}_q$, which in your situation is part of the early work of Steinberg and is included in his 1967-68 Yale lectures. What you need to know is that an absolutely irreducible $FG$-module $M$ over a subfield $F$ of the splitting field $\mathbb{F}_q$ is irreducible over the prime field $\mathbb{F}_p$ if and only if $F$ is the "field of definition" of $M$. Here a field of definition is one generated over the prime field by the traces of representing matrices. (It's probably hard to find an explicit statement and proof of this in the textbooks, but it's implicit.)
In your specific example, it's therefore necessary to be careful about where the traces of the representing matrices lie.