  We calculate the quantum gravitational corrections to the Mukhanov-Sasaki
equation obtained by the canonical quantization of the inflaton-gravity system.
Our approach, which is based on the Born-Oppenheimer decomposition of the
resulting Wheeler-DeWitt equation, was previously applied to a minimally
coupled inflaton. In this article we examine the case of a non minimally
coupled inflaton and, in particular, the induced gravity case is also
discussed. Finally, the equation governing the quantum evolution of the
inflationary perturbations is derived on a de Sitter background. Moreover the
problem of the introduction of time is addressed and a generalized method, with
respect to that used for the minimal coupling case, is illustrated. Such a
generalized method can be applied to the universe wave function when, through
the Born-Oppenheimer factorization, we decompose it into a part which contains
the minisuperspace degrees of freedom and another which describes the
perturbations.
