  We prove that if a quasi-isometry of warped cones is induced by a map between
the base spaces of the cones, the actions must be conjugate by this map. The
converse is false in general, conjugacy of actions is not sufficient for
quasi-isometry of the respective warped cones. For a general quasi-isometry of
warped cones, using the asymptotically faithful covering constructed in a
previous work with Jianchao Wu, we deduce that the two groups are
quasi-isometric after taking Cartesian products with suitable powers of the
integers.
  Secondly, we characterise geometric properties of a group (coarse
embeddability into Banach spaces, asymptotic dimension, property A) by
properties of the warped cone over an action of this group. These results apply
to arbitrary asymptotically faithful coverings, in particular to box spaces. As
an application, we calculate the asymptotic dimension of a warped cone, improve
bounds by Szab\'o, Wu, and Zacharias and by Bartels on the amenability
dimension of actions of virtually nilpotent groups, and give a partial answer
to a question of Willett about dynamic asymptotic dimension.
  In the appendix, we justify optimality of the aforementioned result on
general quasi-isometries by showing that quasi-isometric warped cones need not
come from quasi-isometric groups, contrary to the case of box spaces.
