  We consider the renormalizable $SO(5)/SO(4)$ $\sigma$-model, in which the
Higgs particle has a pseudo-Nambu-Goldstone boson character, and explore what
the minimal field extension required to implement the Peccei-Quinn symmetry
(PQ) is, within the partial compositeness scenario. It turns out that the
minimal model does not require the enlargement of the exotic fermionic sector,
but only the addition of a singlet scalar: it is sufficient that the exotic
fermions involved in partial compositeness and the singlet scalar become
charged under Peccei-Quinn transformations. We explore the phenomenological
predictions for photonic signals in axion searches for all models discussed.
Because of the constraints imposed on the exotic fermion sector by the Standard
Model fermion masses, the expected range of allowed axion-photon couplings
turns out to be generically narrowed with respect to that of standard invisible
axion models, impacting the experimental quest.
