We propose a methodology for simulating attosecond electron dynamics in large molecular systems. Our approach is based on the combination of real time time-dependent-density-functional theory (RT-TDDFT) and polarizable Molecular Mechanics (MMpol) with the point-charge-dipole model of electrostatic induction. We implemented this methodology in the software deMon2k that relies heavily on auxiliary fitted densities. In the context of RT-TDDFT/MMpol simulations, fitted densities allow the cost of the calculations to be reduced drastically on three fronts: (i) the Kohn-Sham potential, (ii) the electric field created by the (fluctuating) electron cloud which is needed in the QM/MM interaction, and (iii) the analysis of the fluctuating electron density on-the-fly. We determine conditions under which fitted densities can be used without jeopardizing the reliability of the simulations. Very encouraging results are found both for stationary and time-dependent calculations. We report absorption spectra of a dye molecule in the gas phase, in nonpolarizable water, and in polarizable water. Finally, we use the method to analyze the distance-dependent response of the environment of a peptide perturbed by an electric field. Different response mechanisms are identified. It is shown that the induction on MM sites allows excess energy to dissipate from the QM region to the environment. In this regard, the first hydration shell plays an essential role in absorbing energy. The methodology presented herein opens the possibility of simulating radiation-induced electronic phenomena in complex and extended molecular systems.