Describe all extension groups of a given subgroup $H \trianglelefteq$ Aff$\mathbb{(F_q)}$ by Aff$\mathbb{(F_q)}/H$

Question

Let $\mathbb{F_q}$ be a finite field. Consider the group Aff$\mathbb{(F_q)}$

Aff$\mathbb{(F_q)} := $ $ \ \begin{Bmatrix} \begin{pmatrix} a&b\\ 0&1\\ \end{pmatrix} \colon a, b \in \mathbb{F_q}, a \neq 0 \end{Bmatrix} $.

Let $H := $ $ \ \begin{Bmatrix} \begin{pmatrix} 1&b\\ 0&1\\ \end{pmatrix} \colon b \in \mathbb{F_q} \end{Bmatrix} $ be a normal subgroup of Aff$\mathbb{(F_q)}$, which is isomorphic to $\mathbb{F_q}$ as groups.

How to describe all extension groups of $H$ by Aff$\mathbb{(F_q)}/H$?

Answer 1

By the Schur-Zassenhaus theorem, every extension $$ 1\rightarrow M\rightarrow E\rightarrow G\rightarrow 1 $$ with $gcd(|M|,|G|)=1$ is split, i.e., $E\cong M\rtimes G$. In other words, we have $H^2(G,M)=1$.

We can apply this here with $G=Aff(q)/H\cong \mathbb{F}_q^*$ of order $q-1$ and $M=H\cong \mathbb{F}_q$ of order $q$, because $gcd(q,q-1)=1$. So there are only split extensions, i.e., semidirect products.