  Given a genus two curve $X: y^2 = x^5 + a x^3 + b x^2 + c x + d$, we give an
explicit parametrization of all other such curves $Y$ with a specified
symplectic isomorphism on three-torsion of Jacobians $\mbox{Jac}(X)[3] \cong
\mbox{Jac}(Y)[3]$. It is known that under certain conditions modularity of $X$
implies modularity of infinitely many of the $Y$, and we explain how our
formulas render this transfer of modularity explicit. Our method centers on the
invariant theory of the complex reflection group $C_3 \times
\operatorname{Sp}_4(\mathbf{F}_3)$. We discuss other examples where complex
reflection groups are related to moduli spaces of curves, and in particular
motivate our main computation with an exposition of the simpler case of the
group $\operatorname{Sp}_2(\mathbf{F}_3) = \mathrm{SL}_2(\mathbf{F}_3)$ and
$3$-torsion on elliptic curves.
