  The minimal $SO(5)/SO(4)$ linear $\sigma$ model is extended including an
additional complex scalar field, singlet under the global $SO(5)$ and the
Standard Model gauge symmetries. The presence of this scalar field creates the
conditions to generate an axion \`a la KSVZ, providing a solution to the strong
CP problem, or an axion-like-particle. Different choices for the PQ charges are
possible and lead to physically distinct Lagrangians. The internal consistency
of each model necessarily requires the study of the scalar potential describing
the $SO(5)\to SO(4)$, electroweak and PQ symmetry breaking. A single minimal
scenario is identified and the associated scalar potential is minimised
including counterterms needed to ensure one-loop renormalizability. In the
allowed parameter space, phenomenological features of the scalar degrees of
freedom, of the exotic fermions and of the axion are illustrated. Two distinct
possibilities for the axion arise: either it is a QCD axion with an associated
scale larger than $\sim10^{5}$ TeV and therefore falling in the category of the
invisible axions; or it is a more massive axion-like-particle, such as a $1$
GeV axion with an associated scale of $\sim200$ TeV, that may show up in
collider searches.
