What is the name of the group linear functions on a finite field? More precisely what is the name for the group
$$\{ X\mapsto \alpha^2X+\beta : \alpha,\beta \in GF(q), \alpha \neq 0\}$$
I've always called it the special affine group, but I see that can mean something else.
 A: Short answer: The Borel subgroup of $\operatorname{PSL}(2,K)$.
Here is a longer answer putting the confused groups into a similar context. I call your group $G$.
As matrices you have $$G = \left\{\begin{bmatrix} \alpha & \beta \\ . & \gamma \end{bmatrix} : \alpha,\beta,\gamma \in K, \alpha\gamma = 1 \right\} \leq B = \left\{\begin{bmatrix} \alpha & \beta \\ . & \gamma \end{bmatrix} : \alpha,\beta,\gamma \in K, \alpha\gamma \neq 0\right\}$$
The right hand side $B$ is a Borel subgroup of $\operatorname{GL}(2,K)$ and the left hand side $G$ is a Borel subgroup of $\operatorname{SL}(2,K)$.
Related group are found by setting $\gamma=1$: $$\operatorname{ASL}(1,K) = \left\{\begin{bmatrix} \alpha & \beta \\ . & 1 \end{bmatrix} : \alpha,\beta \in K, \alpha = 1 \right\} \leq \operatorname{AGL}(1,K) = \left\{\begin{bmatrix} \alpha & \beta \\ . & 1 \end{bmatrix} : \alpha,\beta \in K, \alpha\neq 0\right\}$$
Unfortunately, $\operatorname{ASL}(1,K)$ is not very useful as a name (as it is just the maximal unipotent subgroup of all the groups mentioned in this post, and so has lots of other names, such as $U$).
The first two groups $G$ and $B$ act on the projective line, but they both stabilize $\infty$, and so can be though to act on the affine line, represented by “normalized column vectors” $\bar x = [x,1]$ where $[X,Y] \mapsto [X/Y,1]$ is the normalization (and $\infty$ is the name of the result of normalization when $Y=0$). They act as $$\bar x \mapsto \frac{\alpha}{\gamma} \bar x + \frac{\beta}{\gamma}$$
In the case of $G$, $\gamma = \alpha^{-1}$, so $\frac{\alpha}{\gamma} = \alpha^2$, and I suppose we just need to relabel our $\beta$s. The pointwise stabilizer of the entire line are the centers with $\alpha = \gamma$. Thus your permutation notation applies to the projective linear groups $G/\langle \alpha=\gamma \rangle \leq B/\langle \alpha=\gamma \rangle \leq \operatorname{PGL}(2,K)$.
The nice feature of the second two groups $\operatorname{ASL}(1,K) \leq \operatorname{AGL}(1,K)$ is they act on the affine line without normalization, they send $[x,1]$ directly to $[\alpha x + \beta,1]$ without any need to divide by the second component.
This nice feature is ruined by $\operatorname{ASL}(1,K)$ being sort of a useless name, but for larger matrices, all four groups are useful and distinct.
