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Let $\mathcal{G}$ be a bundle gerbe with connection on a smooth manifold $M$,and let $\rho: G \rightarrow \operatorname{Diff}(M)$ be a smooth action of aFr\'echet--Lie group $G$ on $M$ that preserves the isomorphism class of$\mathcal{G}$. In this setting, we obtain an abelian extension of $G$ thatconsists of pairs $(g,A)$, where $g \in G$, and $A$ is an isomorphism from$\rho_{g}^{*}\mathcal{G}$ to $\mathcal{G}$. We equip this group with a naturalstructure of abelian Fr\'{e}chet--Lie group extension of $G$, under theassumption that the first integral homology of $M$ is finitely generated. As anapplication, we construct the universal central extension (in the category ofFr\'echet--Lie groups) of the group of Hamiltonian diffeomorphisms of asymplectic surface. As an intermediate step, we obtain a central extension ofthe group of exact volume-preserving diffeomorphisms of a 3-manifold whosecorresponding Lie algebra extension is conjectured to be universal.

How an action that stabilizes a bundle gerbe gives rise to a Lie group extension