We give some applications of a Hopf algebra constructed from a group acting on another Hopf algebra A as Hopf automorphisms, namely Molnars smash coproduct Hopf algebra. We find connections between the exponent and Frobenius-Schur indicators of a smash coproduct and the twisted exponents and twisted Frobenius-Schur indicators of the original Hopf algebra A. We study the category of modules of the smash coproduct.
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