  We use cobordism theory to analyse anomalies of finite non-abelian symmetries
in 4 spacetime dimensions. By applying the method of `anomaly interplay', which
uses functoriality of cobordism and naturality of the $\eta$-invariant to
relate anomalies in a group of interest to anomalies in other (finite or
compact Lie) groups, we derive the anomaly for every representation in many
examples motivated by flavour physics, including $S_3$, $A_4$, $Q_8$, and
$\mathrm{SL}(2,\mathbb{F}_3)$. In the case of finite abelian groups, it is well
known that anomalies can be `truncated' in a way that has no effect on
low-energy physics, by means of a group extension. We extend this idea to
non-abelian symmetries. We show, for example, that a system with $A_4$ symmetry
can be rendered anomaly-free, with only one-third as many fermions as na\"ively
required, by passing to a larger symmetry. As another example, we find that a
well-known model of quark and lepton masses utilising the
$\mathrm{SL}(2,\mathbb{F}_3)$ symmetry is anomalous, but that the anomaly can
be cancelled by enlarging the symmetry to a $\mathbb{Z}/3$ extension of
$\mathrm{SL}(2,\mathbb{F}_3)$.
