  It is shown that the Kerr solution exists in the generalized hybrid
metric-Palatini gravity theory and that for certain choices of the function
$f(R,\mathcal R)$ that characterizes the theory, the Kerr solution can be
stable against perturbations on the scalar degree of freedom of the theory. We
start by verifying which are the most general conditions on the function
$f(R,\mathcal R)$ that allow for the general relativistic Kerr solution to also
be a solution of this theory. We perform a scalar perturbation in the trace of
the metric tensor, which in turn imposes a perturbation in both the Ricci and
Palatini scalar curvatures. To first order in the perturbation, the equations
of motion, namely the field equations and the equation that relates the Ricci
and the Palatini curvature scalars, can be rewritten in terms of a fourth-order
wave equation for the perturbation $\delta R$ which can be factorized into two
second-order massive wave equations for the same variable. The usual ansatz and
separation methods are applied and stability bounds on the effective mass of
the Ricci scalar perturbation are obtained. These stability regimes are studied
case by case and specific forms of the function $f(R,\mathcal R)$ that allow
for a stable Kerr solution to exist within the perturbation regime studied are
obtained.
