Let $\phi$ be an automorphizm of $\mathbb{F}_{2^n}$ and $a \in \mathbb{F}_{2^n}$. Let's consider a set of functions $\Gamma L(1,2^n) = \{g_{a,\phi} | a\in\mathbb{F}_{2^n}, \phi \in Aut(\mathbb{F}_{2^n})\}$ such that every $g_{a,\phi} \in \Gamma L(1,2^n)$ acts on elements of $\mathbb{F}_{2^n}$ according to the rule $$ g_{a,\phi} : x \mapsto \varphi(x)a, \text{ } x\in \mathbb{F}_{2^n}.$$ It's easy to see that $g_{a,\phi}$ is linear transformation, so $\Gamma L(1,2^n) \subset GL(n,2)$ and the set $\Gamma L(1,2^n)$ is closed under composition (multiplication) $\Rightarrow$ $\Gamma L(1,2^n)$ is subgroup of $GL(n,2)$ and it's called semilinear group.
I'm trying to describe subgroups $H < GL(n,2)$ such that $\Gamma L(1,2^n) \lneqq H \lneqq GL(n,2) $.
Do somebody know this problem or can give some links on semi-linear groups?
This subgroup is maximal in ${\rm GL}(n,2)$ if and only if $n$ is prime.
If $n=ab$ is composite then $\Gamma {\rm L}(1,2^n) < \Gamma {\rm L}(a,2^b) < {\rm GL}(n,2)$.
I would guess that they are the only such containments except possibly for small exceptions. I checked that it is the only containment when $n=4$.
For the maximality when $n$ is prime see Table 3.5.A of Kleidman and Liebeck "The Subgroups Structure of the Finite Classical Groups" for dimensions $n \ge 12$, and the tables for ${\rm SL}(n,q)$ at the end of Bray, Holt and Roney-Dougal "The Maximal Subgroups of the Low-Dimensional Finite Classical Groups" for $n \le 12$. In general for $n=ab$ with $b$ prime, $\Gamma {\rm L}(a,q^b)$ is maximal in ${\rm GL}(n,q)$. There are a couple of small exceptions with $n=2,3$ but not with $q=2$. These maximal subgroups are in the Aschbacher class ${\mathcal C}_3$ (semilinear groups).