Embedding $\mathbb{G}_a$ into $GL_2$ Let $k$ be an algebraically closed field of characteristic $p$.  I'd like to find interesting examples of closed embeddings $\mathbb{G}_a(k)\hookrightarrow GL_2(k)$, where $\mathbb{G}_a(k)$ is $(k,+)$.  For any non-zero $a\in k$, we have the two standard embeddings:
$$c\longmapsto\begin{pmatrix}1&ac\\0&1\end{pmatrix}\qquad\qquad c\longmapsto\begin{pmatrix}1&0\\ac&1\end{pmatrix}$$
Also, for $p=2$ we have the embedding
$$c\longmapsto\begin{pmatrix}1+ac&ac\\ac&1+ac\end{pmatrix}$$
What are other examples of such embeddings, either in arbitrary positive characteristic, or in a specific positive characteristic?  If you prefer to think in terms of coordinate algebras, this problem is equivalent to finding surjective Hopf algebra maps from $k[GL_2]$ to $k[\mathbb{G}_a]$.  Thanks in advance.
Edit: Thanks to the comments, we have for any $a,b\in k$, not both $0$, an embedding given by
$$c\longmapsto\begin{pmatrix}1-abc&a^2c\\-b^2c&1+abc\end{pmatrix}$$
This example genralizes the previous $3$.  Do all embeddings have this form?
 A: Note that the image of a homomorphism $f:\mathbb{G}_a\to \mathrm{GL}_2$ is smooth, connected, and solvable. Thus, the image is contained in a Borel subgroup of $\mathrm{GL}_2$. But, the Borel subgroups of $\mathrm{GL}_2$ are all conjugate to
$$B_2=\left\{\begin{pmatrix}\ast & \ast\\ 0 & \ast\end{pmatrix}\right\}\subseteq \mathrm{GL}_2$$
Thus, up to conjugation, we may assume that $f$ has image in $B_2$. But, let us note that
$$B_2= R_u(B_2)\rtimes T_2=\left\{\begin{pmatrix}1 & \ast\\ 0 & 1\end{pmatrix}\right\}\rtimes \left\{\begin{pmatrix}\ast & 0\\ 0 & \ast\end{pmatrix}\right\}$$
and so, in particular, we see that $B_2/R_u(B_2)$ is a torus, and thus $f(\mathbb{G}_a)$ lands inside of $R_u(B_2)\cong \mathbb{G}_a$.
The endomorphism ring in characteristic $p$ is $k_\sigma[F]$--the non-commutative $k$-algebra generated by the Frobenius map (e.g. see  [Mil, Example 14.40]). Explicitly if one has $\displaystyle \sum_{i=0}^n a_i F^i$ in $k_\sigma[F]$ this is the map
$$\mathbb{G}_a\to\mathbb{G}_a:x\mapsto \sum_{i=0}^n a_i x^{p^i}$$
Thus, in summation, we see that up to $\mathrm{GL}_2(k)$-conjugacy the homomorphisms $\mathbb{G}_a\to \mathrm{GL}_2$ are all of the form
$$x\mapsto \begin{pmatrix}1 & \displaystyle \sum_{i=0}^n a_i x^{i^p}\\ 0 & 1\end{pmatrix}$$
for some element $\displaystyle \sum_{i=0}^n a_i F^i \in k_\sigma[F]$. Evidently these are embeddings if and only if $n=0$ and $a_0\in k^\times$.
[Mil] Milne, J.S., 2017. Algebraic groups: The theory of group schemes of finite type over a field (Vol. 170). Cambridge University Press.
