Let $p$ be prime and let the elliptic curve $E/\mathbb F_p$ be given an affine equation \begin{equation} y^2 = f(x) = (x - e_1)(x - e_2)(x - e_3), \quad e_i \in \mathbb F_p. \end{equation}
Let the $\mathcal O$ be the unit element of $E(\mathbb F_p)$ and define the map \begin{equation} \varepsilon: E(\mathbb F_p) \to \mathbb (\mathbb F_p^\times/(\mathbb F_p^\times)^2)^3: P \mapsto (\varepsilon_1, \varepsilon_2, \varepsilon_3)(P), \end{equation} where \begin{equation} \varepsilon_i: E(\mathbb F_p) \to \mathbb F_p^\times/(\mathbb F_p^\times)^2: P \mapsto \begin{cases} x - e_i \bmod (\mathbb F_p^\times)^2 & \text{if } \mathcal O \neq P = (x,y) \neq (\theta_i, 0),\newline f'(e_i) \bmod (\mathbb F_p^\times)^2 & \text{if } \mathcal O \neq P = (e_i, 0),\newline 1 \bmod (\mathbb F_p^\times)^2 & \text{if } P = \mathcal O. \end{cases} \end{equation}
I need to show that the image of $\varepsilon$ is \begin{equation} H := \{(1,1,1), (1, a, a), (a, 1, a), (a, a, 1)\}, \quad \text{where} \quad \mathbb F_p^\times/(\mathbb F_p^\times)^2 = \{1, a\} \end{equation} I know that the image of $\varepsilon$ is a subgroup of $H$, and that the kernel of $\varepsilon$ is $2E(\mathbb F_p)$. Therefore, the image of $\varepsilon$ is isomorphic to \begin{equation} E(\mathbb F_p) / 2E(\mathbb F_p). \end{equation}
Because clearly $\#E(\mathbb F_p)[2] \neq 0$, this group cannot be trivial. Hence, at least one nontrivial element of $H$ lies in the image of $\varepsilon$. However, I do not know how to show this implies the other nontrivial elements of $H$ are in the image of $\varepsilon$. Any help would be much appreciated!