$\Pi$-orbits of elliptic curve covers

Let

$$ \mathcal S=(\mathcal I,\Gamma,\Pi) $$

be a seam marked seed built from four compact oriented $2$-dimensional complex orbifold sheets. Each sheet is assumed to be a football type orbifold: its coarse underlying Riemann surface is

$$ |\mathcal O_i|\cong \mathbb P^1 $$

and it has two distinguished cone/orbifold points

$$ p_i^+,\ p_i^- $$

Equivalently, one can think of

$$ \mathcal O_i\cong \mathbb P^1_{(q_i^+,q_i^-)} $$

as a genus zero orbifold curve with two cone points. In the symmetric football case, one may take $q_i^+=q_i^-$.

The sheets are glued or seamed along a distinguished graph $\Gamma$, and $\Pi$ is a finite seed preserving symmetry group acting by piecewise isometries on the seam marked object.

For the purposes of this question, I do not want to get bogged down in the precise construction of $\Pi$. The important point is that $\Pi$ acts on the seam marked seed and therefore moves the marked branch data described below.

Restrict to one orbifold sheet $\mathcal O_i\subset\mathcal I$, and let

$$ \Gamma_i=\Gamma\cap\mathcal O_i $$

be the seam marking on that sheet.

Choose two seam points

$$ a,b\in\Gamma_i $$

and form the branch data

$$ B=\{p_i^+,p_i^-,a,b\} $$

The double cover of the coarse curve

$$ E_B\longrightarrow |\mathcal O_i|\cong\mathbb P^1 $$

branched over $B$ is an elliptic curve. Thus the cone and seam data determine an elliptic modulus.

The group $\Pi$ moves the branch data:

$$ B\mapsto \pi B, \qquad \pi\in\Pi $$

Hence it also moves the associated elliptic curves upstairs:

$$ E_B\mapsto E_{\pi B} $$

So the natural object is not one elliptic curve, but the full $\Pi$-orbit

$$ \{E_{\pi B}:\pi\in\Pi\} $$

Each elliptic curve has a theta function coming from its period lattice. Therefore one can form the $\Pi$-averaged elliptic theta

$$ \Theta_E^\Pi(t)= \frac{1}{|\Pi|} \sum_{\pi\in\Pi} \Theta_{E_{\pi B}}(t) $$

Question. I have seen, in the arithmetic study of elliptic curves over finite fields, a healthy dosage of internal symmetry groups (of elliptic curves) such as automorphism groups/torsion subgroups. My question is very different from that. It asks about studying $\Pi$-orbits of elliptic curve covers through their theta averages based over $\mathcal S$. Is there any consideration of this more external symmetry method of analyzing elliptic curves?