  We give some applications of a Hopf algebra constructed from a group acting
on another Hopf algebra A as Hopf automorphisms, namely Molnar's 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.
