  Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable
surface of genus $g \geq 2$. In this article, we derive necessary and
sufficient conditions under which two torsion elements in $\mathrm{Mod}(S_g)$
will have conjugates that generate a finite symmetric or an alternating
subgroup of $\mathrm{Mod}(S_g)$. Furthermore, we characterize when an
involution would lift under the branched cover induced by an alternating action
on $S_g$. Moreover, up to conjugacy, we derive conditions under which a given
periodic mapping class is contained in a symmetric or an alternating subgroup
of $\mathrm{Mod}(S_g)$. In particular, we show that symmetric or alternating
subgroups can not contain irreducible mapping classes and hyperelliptic
involutions. Finally, we classify the symmetric and alternating actions on
$S_{10}$ and $S_{11}$ up to a certain equivalence we call weak conjugacy.
