  We consider an augmented Einstein-Maxwell-scalar model including an
axionic-type coupling between the scalar and electromagnetic field. We study
dyonic black hole solutions in this model. For the canonical axionic coupling
emerging from high energy physics, all charged black holes have axion hair. We
present their domain of existence and investigate some physical properties. For
other axionic-type couplings, two classes of black hole solutions may co-exist
in the model: scalar-free Reissner-Nordstr\"om black holes and scalarised black
holes. We show that in some region of the parameter space, the scalar-free
solutions are unstable. Then, there is non-uniqueness since new scalarised
black hole solutions with the same global charges also exist, which are
entropically preferred over the scalar-free solutions and, moreover, emerge
dynamically from the instability of the former.
