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In Weinberg QFT Vol.1, it says one can enlarge the symmetry group $H$ to the universal covering group $C$ such that one obtains a trivial cocycle or $C$ is simply connected whereas $H$ is not. I get the mathematical meaning of the statement. I wonder whether this also requires $C$ being an extended symmetry of the system somehow though I thought $H$ is the largest symmetry group of the system. I am kind of confused here how can one further extend the system invariant under the action of the universal covering group unless one define the action of elements beyond $H$ being trivial.