The reconstruction of a (smooth, plane) quartic from its bitangents is a classical problem. S. Aronhold [see A. B. Coble, “Algebraic geometry and theta functions”, section 43, American Mathematical Society Colloquium Publications 10 (1929; JFM 55.0808.02)] proved that the reconstruction is possible (and explicit) given a particular set of seven bitangents called ‘Aronhold system’. Knowing this set is the same as knowing the \(28\) bitangents with a symplectic structure (induced by the action of the Weil pairing on \(2\)-torsion points of the Jacobian). Thus the problem reduces to prove that the symplectic structure is uniquely determined by the set of bitangents alone. This is carried out by the author in this article using the classification of \(2\)-transitive groups and the Coble-Recillas construction [cf. S. Recillas, Bol. Soc. Mat. Mex., II. Ser. 19, 9–13 (1974; Zbl 0343.14012)].
Note that similar results have been obtained in the case of generic curves of genus \(g \geq 3\) by L. Caporaso and E. Sernesi [J. Algebr. Geom. 12, 225–244 (2003; Zbl 1080.14523) and J. Reine Angew. Math. 562, 101–135 (2003; Zbl 1039.14011)].