The time-integrated untagged Dalitz plot of the three-body hadronic charmless decay ${\overline{B}}^{0}\ensuremath{\rightarrow}{K}_{S}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ is studied using a $pp$ collision data sample recorded with the LHCb detector, corresponding to an integrated luminosity of $3.0\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. The decay amplitude is described with an isobar model. Relative contributions of the isobar amplitudes to the ${\overline{B}}^{0}\ensuremath{\rightarrow}{K}_{S}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decay branching fraction and $CP$ asymmetries of the flavor-specific amplitudes are measured. The $CP$ asymmetry between the conjugate ${\overline{B}}^{0}\ensuremath{\rightarrow}{K}^{*}(892{)}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}$ and ${B}^{0}\ensuremath{\rightarrow}{K}^{*}(892{)}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decay rates is determined to be $\ensuremath{-}0.308\ifmmode\pm\else\textpm\fi{}0.062$.