  We show that semisimple Hopf algebras having a self-dual faithful irreducible
comodule of dimension 2 are always obtained as abelian extensions with quotient
Z_2. We prove that nontrivial Hopf algebras arising in this way can be regarded
as deformations of binary polyhedral groups and describe its category of
representations. We also prove a strengthening of a result of Nichols and
Richmond on cosemisimple Hopf algebras with a 2-dimensional irreducible
comodule in the finite dimensional context. Finally, we give some applications
to the classification of certain classes of semisimple Hopf algebras.
