  Quantum-correlated $D\bar{D}$ pairs collected by the BESIII experiment at the
$\psi(3770)$ resonance, corresponding to an integrated luminosity of 2.93
fb$^{-1}$, are used to study the $D^0 \rightarrow K^{0}_S\pi^{+} \pi^{-}
\pi^{0}$ decay mode. The $C\!P$-even fraction of $D^0 \rightarrow
K^{0}_S\pi^{+} \pi^{-} \pi^{0}$ decays is determined to be $0.235\pm 0.010\pm
0.002$, where the first uncertainty is statistical and the second is
systematic.
