I am looking to classify the homomorphisms from the group $PSL(2,p)=Aut(\mathbb{P}^1\mathbb{F}_p)$ to $PGL(n,2)=GL(n,2)=Aut(\mathbb{F}_{2^n})$ when $p$ is a Mersenne prime, i.e. $2^n=p+1$.

There is a bijection $\mathbb{P}^1\mathbb{F}_p \rightarrow \mathbb{F}_{2^n} \cong \mathbb{F}_2[x]/g(x)$ given by $k \mapsto x^\infty+x^k$ where $x^\infty=0$. Given a Möbius transformation $f$, define the homomorphism $T$ by $T_f(x^\infty+x^k)=x^{f(\infty)}+x^{f(k)}$.

When $n=3$, $p=7$, $T$ is an isomorphism. What is $im(T)\subset PGL(n,2)$?

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\GL{GL}$I am looking to classify the homomorphisms from the group $\PSL(2,p)=\Aut(\mathbb{P}^1\mathbb{F}_p)$ to $\PGL(n,2)=\GL(n,2)=\Aut(\mathbb{F}_{2^n})$ when $p$ is a Mersenne prime, i.e. $2^n=p+1$.

There is a bijection $\mathbb{P}^1\mathbb{F}_p \rightarrow \mathbb{F}_{2^n} \cong \mathbb{F}_2[x]/g(x)$ given by $k \mapsto x^\infty+x^k$ where $x^\infty=0$. Given a Möbius transformation $f$, define the homomorphism $T$ by $T_f(x^\infty+x^k)=x^{f(\infty)}+x^{f(k)}$.

When $n=3$, $p=7$, $T$ is an isomorphism (see Brown and Loehr - Why is $\PSL(2, 7) \cong \GL(3, 2)$?). What is $\operatorname{im}(T)\subset \PGL(n,2)$?