Integrability criterion for abelian extensions of Lie groups

We establish a criterion for when an abelian extension of infinite-dimensional Lie algebras $\mathfrak{\hat{g}} = \mathfrak{g} \oplus_\omega \mathfrak{a}$ integrates to a corresponding Lie group extension $A \hookrightarrow \widehat{G} \twoheadrightarrow G$, where $G$ is connected, simply connected and $A \cong \mathfrak{a} \slash \Gamma$ for some discrete subgroup $\Gamma \subseteq \mathfrak{a}$. When $\pi_1(G)\neq 0$, the kernel $A$ is replaced by a central extension $\widehat{A}$ of $\pi_1(G)$ by $A$.