  For a 3-manifold M, the twist group Twist(M) is the subgroup of the mapping
class group Mod(M) generated by twists about embedded 2-spheres. We study the
Nielsen realization problem for subgroups of Twist(M). We prove that a
nontrivial subgroup G<Twist(M) is realized by diffeomorphisms if and only if G
is cyclic and M is a connected sum of lens spaces. We also apply our methods to
the Burnside problem for 3-manifolds and show that Diff(M) does not contain an
infinite torsion group when M is reducible and not a connected sum of lens
spaces.
