  We obtain a partial classification of the finite groups $G$ for which the
integral group ring $\mathbb{Z} G$ has projective cancellation, i.e. for which
$P \oplus \mathbb{Z} G \cong Q \oplus \mathbb{Z} G$ implies $P \cong Q$ for
projective $\mathbb{Z} G$-modules $P$ and $Q$. To do this, we prove a
cancellation theorem based on a relative version of the Eichler condition and
use it to establish the cancellation property for a large class of finite
groups. We then use a group theoretic argument to precisely determine the class
of groups not covered by this result. The final classification is then obtained
by applying results of Swan, Chen and Bley-Hofmann-Johnston which show failure
of the cancellation property for certain groups.
