In regular group cohomology theory, it is well known that $\mathrm{H^2(G,A)} \cong \mathrm{Ext(G,A)}$, where $\mathrm{Ext(G,A)}$ denotes the class of all groups $\mathrm{U}$ which make the following sequence exact: $$\mathrm{0 \rightarrow A \rightarrow U \rightarrow G \rightarrow 1}$$ and the action of $\mathrm{G}$ on $\mathrm{A}$, is the same as the conjugation by the corresponding element from $\mathrm{U}$.
What about the case when $\mathrm{G}$ is a pro-finite group and one considers $\mathrm{H_{cont}^2(G,A)}$ instead of the regular cohomology? Is there a similiar interpretation to the above isomorphism, say, for example, the class of groups $\mathrm{U}$ with the same property but with continous arrows? In other words, may one writes something like that $\mathrm{H_{cont}^2(G,A)} \cong \mathrm{Ext_{cont}(G,A)}$ (If the right object really exist)?
I am familiar with the relation to Brauer groups, but I like to work with group extensions.
Thank you!