This paper investigates ways to enlarge the
Hamiltonian subgroup {\rm Ham} of the symplectomorphism group {\rm Symp} of a
symplectic manifold $(M, \omega)$ to a group that both intersects every connected
component of {\rm Symp} and characterizes symplectic bundles with fiber $M$ and
closed connection form. As a consequence, it is shown that bundles with closed
connection form are stable under appropriate small perturbations of the symplectic
form. Further, the manifold $(M,\omega)$ has the property that every symplectic
$M$-bundle has a closed connection form if and only if the flux group vanishes and
the flux homomorphism extends to a crossed homomorphism defined on the whole group
{\rm Symp}. The latter condition is equivalent to saying that a connected component
of the commutator subgroup [{\rm Symp}, {\rm Symp}] intersects the identity
component of {\rm Symp} only if it also intersects {\rm Ham}. It is not yet clear
when this condition is satisfied. We show that if the symplectic form vanishes on
2-tori, the flux homomorphism extends to the subgroup of {\rm Symp} acting trivially
on $\pi_1(M)$. We also give an explicit formula for the Kotschick--Morita extension
of the flux homomorphism in the monotone case. The results in this paper belong to
the realm of soft symplectic topology, but raise some questions that may need hard
methods to answer.