  We consider generalized Haagerup categories such that $1 \oplus X$ admits a
$Q$-system for every non-invertible simple object $X$. We show that in such a
category, the group of order two invertible objects has size at most four. We
describe the simple objects of the Drinfeld center and give partial formulas
for the modular data. We compute the remaining corner of the modular data for
several examples and make conjectures about the general case. We also consider
several types of equivariantizations and de-equivariantizations of generalized
Haagerup categories and describe their Drinfeld centers.
  In particular, we compute the modular data for the Drinfeld centers of a
number of examples of fusion categories arising in the classification of
small-index subfactors: the Asaeda-Haagerup subfactor; the $3^{\Z_4} $ and
$3^{\Z_2 \times \Z_2} $ subfactors; the $2D2$ subfactor; and the $4442$
subfactor.
  The results suggest the possibility of several new infinite families of
quadratic categories. A description and generalization of the modular data
associated to these families in terms of pairs of metric groups is taken up in
the accompanying paper \cite{GI19_2}.
