  We classify twisted conjugacy classes of type D associated to the sporadic
simple groups. This is an important step in the program of the classification
of finite-dimensional pointed Hopf algebras with non-abelian coradical. As a
by-product we prove that every complex finite-dimensional pointed Hopf algebra
over the group of automorphisms of M12, J2, Suz, He, HN, T is the group
algebra. In the appendix we improve the study of conjugacy classes of type D in
sporadic simple groups.
