Let M=(M,Ο) be either the product S2ΓS2 or the non-trivial
S2 bundle over S2 endowed with any symplectic form Ο. Suppose a
finite cyclic group Znβ is acting effectively on (M,Ο) through
Hamiltonian diffeomorphisms, that is, there is an injective homomorphism
ZnββͺHam(M,Ο). In this paper, we investigate the homotopy
type of the group SympZnβ(M,Ο) of equivariant symplectomorphisms. We
prove that for some infinite families of Znβ actions satisfying certain
inequalities involving the order n and the symplectic cohomology class
[Ο], the actions extends to either one or two toric actions, and
accordingly, that the centralizers are homotopically equivalent to either a
finite dimensional Lie group, or to the homotopy pushout of two tori along a
circle. Our results rely on J-holomorphic techniques, on Delzant's
classification of toric actions, on Karshon's classification of Hamiltonian
circle actions on 4-manifolds, and on the Chen-Wilczy\'nski classification of
smooth Znβ-actions on Hirzebruch surfaces.Comment: 36 pages. Initial release. Comments welcom