  We prove that the Waldhausen nilpotent class group of an injective index 2
amalgamated free product is isomorphic to the Farrell-Bass nilpotent class
group of a twisted polynomial extension. As an application, we show that the
Farrell-Jones Conjecture in algebraic K-theory can be sharpened from the family
of virtually cyclic subgroups to the family of finite-by-cyclic subgroups.
