  For any symplectic form $\omega$ on $T^2\times S^2$ we construct infinitely
many nonisomorphic finite groups which admit effective smooth actions on
$T^2\times S^2$ that are trivial in cohomology but which do not admit any
effective symplectic action on $(T^2\times S^2,\omega)$. We also prove that for
any $\omega$ there is another symplectic form $\omega'$ on $T^2\times S^2$ and
a finite group acting symplectically and effectively on $(T^2\times
S^2,\omega')$ which does not admit any effective symplectic action on
$(T^2\times S^2,\omega)$.
  A basic ingredient in our arguments is the study of the Jordan property of
the symplectomorphism groups of $T^2\times S^2$. A group $G$ is Jordan if there
exists a constant $C$ such that any finite subgroup $\Gamma$ of $G$ contains an
abelian subgroup whose index in $\Gamma$ is at most $C$. Csik\'os, Pyber and
Szab\'o proved recently that the diffeomorphism group of $T^2\times S^2$ is not
Jordan. We prove that, in contrast, for any symplectic form $\omega$ on
$T^2\times S^2$ the group of symplectomorphisms $Symp(T^2\times S^2,\omega)$ is
Jordan. We also give upper and lower bounds for the optimal value of the
constant $C$ in Jordan's property for $Symp(T^2\times S^2,\omega)$ depending on
the cohomology class represented by $\omega$. Our bounds are sharp for a large
class of symplectic forms on $T^2\times S^2$.
