  In this note we determine, for an arbitrary but a fixed degree $d$, an
algorithm to list the possible values $m$ for which $M_g^{Pl}(\mathbb{Z}/m)$ is
non-empty where $\mathbb{Z}/m$ denotes the cyclic group of order $m$. In
particular, we prove that $m$ should divide one of the integers: $d-1$, $d$,
$d^2-3d+3$, $(d-1)^2$, $d(d-2)$ or $d(d-1)$. Secondly, consider a curve
$\delta\in M_g^{Pl}$ with $g=(d-1)(d-2)/2$ such that $Aut(\delta)$ has an
element of "very large" order, in the sense that this element is of order
$d^2-3d+3$, $(d-1)^2$, $d(d-2)$ or $d(d-1)$. Then we investigate the groups $G$
for which $\delta\in\widetilde{M_g^{Pl}(G)}$ and also we determine the locus
$\widetilde{M_g^{Pl}(G)}$ in these situations. Moreover, we work with the same
question when $Aut(\delta)$ has an element of "large" order $\ell d$, $\ell
(d-1)$ or $\ell(d-2)$ with $\ell\geq 2$ an integer.
