  We examine the contribution of small instantons to the axion mass in various
UV completions of QCD. We show that the reason behind the potential dominance
of such contributions is the non-trivial embedding of QCD into the UV theory.
The effects from instantons in the partially broken gauge group appear as
"fractional instanton" corrections in the effective theory. These will exhibit
unusual dependences on the various scales in the problem whenever the index of
embedding is non-trivial. We present a full one-instanton calculation of the
axion mass in the simplest product group models, carefully keeping track of
numerical prefactors. Rather than using a 't Hooft operator approximation we
directly evaluate the contributions to the vacuum bubble, automatically
capturing the effects of closing up external fermion lines with Higgs loops.
This approach is manifestly finite and removes the uncertainty associated with
introducing a cutoff scale for the Higgs loops. We verify that the small
instantons may dominate over the QCD contribution for very high breaking scales
and at least three group factors.
