  We provide the converses to two results of J. Roe (Geom. Topol. 2005): first,
the warped cone associated to a free action of an a-T-menable group admits a
fibred coarse embedding into a Hilbert space, and second, a free action
yielding a warped cone with property A must be amenable. We construct examples
showing that in both cases the freeness assumption is necessary. The first
equivalence is obtained also for other classes of Banach spaces, in particular
for $L^p$-spaces.
