Let
M
=
(
M
,
ω
)
M=(M,\omega )
be either
S
2
×
S
2
S^2 \times S^2
or
C
P
2
#
C
P
2
¯
\mathbb {C}P^2\# \overline {\mathbb {C}P^2}
endowed with any symplectic form
ω
\omega
. Suppose a finite cyclic group
Z
n
\mathbb {Z}_n
is acting effectively on
(
M
,
ω
)
(M,\omega )
through Hamiltonian diffeomorphisms, that is, there is an injective homomorphism
Z
n
↪
H
a
m
(
M
,
ω
)
\mathbb {Z}_n\hookrightarrow Ham(M,\omega )
. In this paper, we investigate the homotopy type of the group
S
y
m
p
Z
n
(
M
,
ω
)
Symp^{\mathbb {Z}_n}(M,\omega )
of equivariant symplectomorphisms. We prove that for some infinite families of
Z
n
\mathbb {Z}_n
actions satisfying certain inequalities involving the order
n
n
and the symplectic cohomology class
[
ω
]
[\omega ]
, the actions extend 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
J
-holomorphic techniques, on Delzant’s classification of toric actions, on Karshon’s classification of Hamiltonian circle actions on
4
4
-manifolds, and on the Chen-Wilczyński classification of smooth
Z
n
\mathbb {Z}_n
-actions on Hirzebruch surfaces.