  This paper discusses topological and locally linear actions of finite groups
on $S^4$. Local linearity of the orientation preserving actions on $S^4$ forces
the group to be a subgroup of $SO(5)$. On the other hand, orientation reversing
topological actions of "exotic" groups $G$ (i.e. $G\not\subset O(5)$) on $S^4$
are constructed, and local linearity and stable smoothability of the actions
are studied.
