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I felt that the earlier question may be too challenging, so let me provide a different angle and more infos to tackle an easier and separate problem. Let us consider a more explicit a short exact ...

Let us consider a short exact sequence: $$ 1\to N\to G\to Q \to 1, $$ where $N$, $Q$, and $G$ can be continuous Lie groups in general (or finite groups). Suppose I have the data and the computations ...