Second Lie group cohomology: e-smooth cocycles

Question

It is well known that in the category of groups, we have: \begin{equation*} Ext(G,A) \cong H^2_{gr}(G,A), \end{equation*} where by $H^2_{gr}(G,A)$ I mean the second group cohomology: \begin{equation*} H^2_{gr}(G,A):=\{\omega: G \times G \to A| \omega \; \; \text{is a cocycle} \}/\sim. \end{equation*} If we now turn to Lie groups, and we define the notion of central extension of a connected Lie group $G$ by a connected abelian Lie group $A$, $H^2_{gr}(G,A)$ does not classify all central extensions in the Lie theoretic sense. My naive approach would be to consider only smooth cocycles and define \begin{equation*} H^2_{s,gr}(G,A):=\{\omega: G \times G \to A| \omega \; \; \text{is a cocycle and it is smooth}\}/ \sim. \end{equation*} Intuitively, I would say that $Ext_{Lie}(G,A) \cong H^2_{s,gr}(G,A)$. However, in this article it is stated that $H^2_{s,gr}(G,A)$ only classifies those central extensions that admit a global smooth section. I do not see why this would be the case. Furthermore, they claim that the cocycles, which are smooth around the identity \begin{equation*} H^2_{es,gr}(G,A):=\{\omega: G \times G \to A| \omega \; \; \text{is a cocycle, smooth around} \; \; e \in G\}/ \sim. \end{equation*} classify all central extensions of $G$ by $A$ in the Lie theoretic sense, i.e. $Ext_{Lie}(G,A) \cong H^2_{es,gr}(G,A)$, this is proposition 3.11 in the paper. They provide a sketch of a proof, which I can not follow. Could someone help me to write down this group isomorphism quite explicitly and to see why it is an isomorphism?