Let $E/k$ be an elliptic curve over some algebraically closed field $k$ of characteristic $p\ge 0$. It's known that $Aut(E)$ acts faithfully on the Tate module $T_\ell(E)$ ($\ell\ne p$) with determinant 1. Is there a complete description of the actions of $Aut(E)$ on $T_\ell(E)$? Ie, for any elliptic curve as above, can we describe the subgroup $Aut(E)\subset SL_2(\mathbb{Z}_\ell)\subset Aut(T_\ell(E))$ up to conjugacy?

In characteristic 0 the action can be computed analytically and we find that the "extra automorphisms" $i,\rho$ of orders 4,6 (corresponding to $j$-invariant 1728,0) essentially act via conjugates of

$$M_i =\begin{bmatrix}0 & 1\\-1 & 0\end{bmatrix}\qquad M_\rho = \begin{bmatrix}1 & 1 \\ -1 & 0\end{bmatrix}$$

Reducing mod $p$ one finds that the reductions $\overline{i},\overline{\rho}$ act on the $\ell$-power torsion in the same way, and so the matrices are the same.

Thus, my question reduces to: If $char(k) = 2$ or 3, and $j = 0\equiv 1728$, then we have automorphism groups of order either 12 (characteristic 3), or 24 (characteristic 2).

In this case what is the subgroup $Aut(E)\subset SL_2(\mathbb{Z}_\ell)$? We certainly get both the automorphisms $\overline{i},\overline{\rho}$ with matrices $M_i,M_\rho$, but there isn't necessarily a *single* choice of basis for $T_\ell(E)$ such that $\overline{i},\overline{\rho}$ have matrices $M_i,M_\rho$ respectively. Also, in characteristic 2, there are additional automorphisms which aren't in the group generated by $\overline{i},\overline{\rho}$.

I feel like this must have been done somewhere, but I can't find any references for this.