It is natural to ask whether an action of a finite cyclic group extends to a circle action. Here, the action is on a symplectic manifold of dimension four. Admitting a circle action implies that a simply connected closed symplectic four-manifold is either the projective plane or obtained from an $S^2$ bundle over $S^2$ by $k$ blowups. I will show that for $k$ small enough, any cyclic action that is trivial on homology extends to a circle action, and present a case in which the action does not extend. I will also discuss how we approach this question for a general $k$. The proofs combine holomorphic and combinatorial methods. The talk is based on a joint work with River Chiang.