This is mainly a request for a straightforward reference (preferably at textbook level). The question comes up while responding to a question raised by non-specialists in finite group representations.

Let $G$ be a finite group and let $F \subset E$ be finite fields, say with $E$ a splitting field for $G$ but perhaps $F$ not. Starting with a simple module $M$ of dimension $d$ for the group algebra $E[G]$, consider the $F[G]$-module $N$ obtained by "restriction of scalars" from $M$: here $M$ as a vector space over $F$ typically has larger dimension related to $d$ while the representing matrices for $G$ over $F$ involve a rewriting of the matrices over $E$. While this is elementary in principle, it's a bit complicated to write down in practice. In the process, one could chart a precise relationship between simple modules over the field $F$ (maybe not a splitting field) and those over $E$.

Is this written down clearly for non-specialists?

Going the other way, from a module over an arbitrary field $F$ to one over a splitting field $E$, is a more standard topic in textbooks about which much can be said in prime characteristic as well as in characteristic 0 (where the Schur index becomes a leading idea). See for example the extensive discussion in Chapter 9 "Changing the field" in *Character Theory of Finite Groups* by I.M. Isaacs along with the following chapter. But I don't recall seeing comparable detail on restriction of scalars. The question I've been discussing with colleagues isn't especially deep or advanced, but a clear reference would help.

ADDED: To make the situation more precise, it may be enough (following Geoff) to assume that $F$ is not a splitting field for the given simple $E[G]$-module $M$ while $E$ is a minimal splitting field for it. (Or you might just take $E$ to be a minimal splitting field for $G$.) Then if $[E:F] = n$, the restricted $F[G]$-module $N$ has dimension $nd$ over $F$. In this situation one expects $N$ to be simple for the smaller group algebra, which in turn yields an $nd$-dimensional module over $E$ after tensoring, etc. The fields involved being finite, there is no Schur index complication and the Galois group of $E/F$ is easy to work with. So the matrix version over $F$ of the given representation over $E$ should be readily described in block form. With luck, this is all I've been asked to explain in special cases, but it would be reassuring to see a concise published version which I couldn't readily extract from books I've seen.