Let $p$ a prime and $n$ a positive integer, What are the outer automorphisms of the finite linear group $GL_2(\mathbb{Z}/p^{n}\mathbb{Z})$? do we know a complete list of them? Is there any thing on the literature about this?
After some search in literature, I found some types of automorphisms which are not inner, namely the following:
- Automorphisms induced by automorphisms of the ring $R$ (This is just the identity in our case $R = \mathbb{Z}/p^{n}\mathbb{Z}$)
- Composition of the inverse with the transpose: $T\mapsto (T^{t})^{-1}$
- Maps of the form $T\mapsto \chi(T)T$, where $\chi$ is a group homomorphism from $GL_2(\mathbb{Z}/p^{n}\mathbb{Z})$ to the center $\{\lambda I : \lambda\in (\mathbb{Z}/p^{n}\mathbb{Z})^{\times}\}$ which satisfy $\chi(Z)\neq Z^{-1}$ for all $Z\neq I$
These types of automorphisms were called standard in literature, does this indicates the possible existence of no standard ones? Any ways I could not find a complete classification for the case I am interested on.