An amplitude analysis of the KSKS system produced in radiative J/ψ decays is performed using the (1310.6±7.0)×106 J/ψ decays collected by the BESIII detector. Two approaches are presented. A mass-dependent analysis is performed by parametrizing the KSKS invariant mass spectrum as a sum of Breit-Wigner line shapes. Additionally, a mass-independent analysis is performed to extract a piecewise function that describes the dynamics of the KSKS system while making minimal assumptions about the properties and number of poles in the amplitude. The dominant amplitudes in the mass-dependent analysis include the f0(1710), f0(2200), and f2′(1525). The mass-independent results, which are made available as input for further studies, are consistent with those of the mass-dependent analysis and are useful for a systematic study of hadronic interactions. The branching fraction of radiative J/ψ decays to KSKS is measured to be (8.1±0.4)×10-4, where the uncertainty is systematic and the statistical uncertainty is negligible.