  We demonstrate the existence of numerous non-spin 4-manifolds for which the
smooth Nielsen realization problem fails; namely, there exist finite subgroups
of their mapping class groups that cannot be realized by any group of
diffeomorphisms. This extends and complements recent results for spin
4-manifolds. Our examples span virtually all possible intersection forms, both
even and odd, indefinite and definite, and include many irreducible
4-manifolds. To derive these examples, we study multi-twists, projective
twists, and multi-reflections, which are all mapping classes supported around
collections of embedded spheres and projective planes. Our obstructions to
Nielsen realization are based on the work of Konno. We investigate projective
twists in further detail, and notably, employ them to show that, for many
closed symplectic 4-manifolds, the symplectic Torelli group is not generated by
squared Dehn twists.
