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Thank you for the answer. I don't see any contradiction when you said that $\hat{L}_n$ is locally compact: it does not prevent the group from being a Polish group generated by a compact subset, as you stated.
@YCor Thank you for a comment. In your example, $H$ is abelian and the action is non-trivial, while my assumption says the action is trivial (since $H/[H,H] = H$); so it's not a counter example.
