  In the present paper we study the scalar sector of the quasinormal modes of
charged general relativistic, static and spherically symmetric black holes
coupled to nonlinear electrodynamics and embedded in a class of scalar-tensor
theories. We find that for certain domain of the parametric space there exist
unstable quasinormal modes. The presence of instabilities implies the existence
of scalar-tensor black holes with primary hair that bifurcate from the embedded
general relativistic black-hole solutions at critical values of the parameters
corresponding to the static zero-modes. We prove that such scalar-tensor black
holes really exist by solving the full system of scalar-tensor field equations
for the static, spherically symmetric case. The obtained solutions for the
hairy black holes are non-unique and they are in one to one correspondence with
the bounded states of the potential governing the linear perturbations of the
scalar field. The stability of the non-unique hairy black holes is also
examined and we find that the solutions for which the scalar field has zeros
are unstable against radial perturbations. The paper ends with a discussion on
possible formulations of a new classification conjecture.
