  We characterize, for every higher smooth stack equipped with "tangential
structure", the induced higher group extension of the geometric realization of
its higher automorphism stack. We show that when restricted to smooth manifolds
equipped with higher degree topological structures, this produces higher
extensions of homotopy types of diffeomorphism groups. Passing to the groups of
connected components, we obtain abelian extensions of mapping class groups and
we derive sufficient conditions for these being central. We show as a special
case that this provides an elegant re-construction of Segal's approach to
$\mathbb{Z}$-extensions of mapping class groups of surfaces that provides the
anomaly cancellation of the modular functor in Chern-Simons theory. Our
construction generalizes Segal's approach to higher central extensions of
mapping class groups of higher dimensional manifolds with higher tangential
structures, expected to provide the analogous anomaly cancellation for higher
dimensional TQFTs.
