  We prove the fibred Farrell--Jones Conjecture (FJC) in $A$-, $K$-, and
$L$-theory for a large class of suspensions of relatively hyperbolic groups, as
well as for all suspensions of one-ended hyperbolic groups. We deduce two
applications:
  (1) FJC for the automorphism group of a one-ended group hyperbolic relative
to virtually polycyclic subgroups;
  (2) FJC is closed under extensions of FJC groups with kernel in a large class
of relatively hyperbolic groups.
  Along the way we prove a number of results about JSJ decompositions of
relatively hyperbolic groups which may be of independent interest.
