  Henn and Komiya-Kuribayashi listed, independently, the groups $G$ for which
$\widetilde{M_3^{Pl}(G)}$ is non-empty. In this paper, we determine the loci
$\widetilde{M_6^{Pl}(G)}$, corresponding to non-singular degree $5$ projective
plane curves, which are non-empty. Also, we present the analogy of Henn's
results for quartic curves concerning non-singular plane model equations
associated to these loci.
