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Let $G$ be a finite non-abelian simple group, $M.G$ the Schur covering group of $G$. Is there a method to find the order of a preimage of an element of order 2 in the natural homomorphism $\pi: M.G\rightarrow G$?

We can find the result using the 'Atlas of finite groups' for some finite simple groups, but is there a general method?

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