How to find out if a semilinear representation is irreducible (possibly with gap)

I will use the notation $\mathbb{F}_q$ for the finite field with $q$ elements.

Even ignoring $\mathrm{Gal}(\mathbb{F}_{q^f}\mid\mathbb{F}_q)$, many subgroups of $\mathbb{F}_{q^f}^\times$ act irreducibly on $\mathbb{F}_{q^f}$ as a $\mathbb{F}_q$-vector space. Indeed, $\mathbb{F}_{q^f}$ is an irrep of $\langle\omega\rangle$ iff it is a simple $\mathbb{F}_q(\omega)$-module, which is true iff $\mathbb{F}_q(\omega)=\mathbb{F}_{q^f}$, which is in turn true iff $\omega$ is not in any proper subfield. In the case of $q^f=2^6$, we can verify a $9$th root of unity is not contained in the subfields of sizes $2^2$ or $2^3$ because its order does not divide $2^2-1$ or $2^3-1$.

I'd be curious which subgroups of the semilinear group act irreducibly in general.

Dunno how to help with GAP unfortunately, sorry.