Consider a genus $2$ curve $C$ and its Jacobian $J$ (for simplicity, over the field $\overline{\mathbb{Q}}$ or $\mathbb{C}$). Assume that $J$ is $(2,2)$-isogenous to the direct square $E^2$ of an elliptic curve $E$. In other words, there are two complementary quadratic covers $\varphi: C \to E$ and $\psi: C \to E$. It is not a secret that one can explicitly write out formulas of $\varphi$, $\psi$.
It is known that the endomorphism ring of $E^2$ is nothing but the matrix ring $\mathrm{Mat}_2(O)$, where $O := \mathrm{End}(E)$. I suppose that $O$ is the maximal order in the field $K := O \otimes \mathbb{Q}$. Thereby, $\mathrm{Mat}_2(O)$ is a maximal order in $\mathrm{Mat}_2(K) \simeq \mathrm{End}(J) \otimes \mathbb{Q}$. Denote by $G := \mathrm{Aut}(C)$ the automorphism group of $C$. Obviously, $G$ also acts on $J$.
Is there a quite simple way to explicitly find the representation $\varrho: G \to \mathrm{Mat}_2(K)$? Am I right that $\mathrm{Im}(\varrho) \subset \mathrm{Mat}_2(O)$?
I am mainly interested in the Bolza curve $C: y^2 = x^5 + x$ for which the $j$-invariant of $E$ is equal to $8000$, i.e., $O \simeq \mathbb{Z}[\sqrt{-2}]$. Besides, $G \simeq \mathrm{GL}_2(\mathbb{F}_3)$ is a group of order $48$, which is maximal among automorphism groups of genus $2$ curves. The Bolza curve is a classical object, hence someone probably already computed the representation $\varrho$. If so, I will be very grateful for a reference.