Following our preceding research [P. M. McLaurin, R. Merritt, J. C. Domínguez, E. S. Teixeira and J. A. Morales, Phys. Chem. Chem. Phys., 2019, 21, 5006], we present an electron nuclear dynamics (END) investigation of H+ + H2O at ELab = 28.5-200.0 eV in conjunction with a computational procedure to induce symmetry breaking during evolution. The investigated system is a computationally feasible prototype to simulate water radiolysis reactions in ion cancer therapy. END is a time-dependent, variational, non-adiabatic, and on-the-fly method, which utilizes classical mechanics for nuclei and a Thouless single-determinantal state for electrons. In this study, a procedure inherent to END introduces low degrees of symmetry breaking into the reactants' restricted Hartree-Fock (RHF) state to induce a higher symmetry breaking during evolution. Specifically, the Thouless exponential operator acting on the RHF reference generates an axial spin density wave (ASDW) state according to Fukutome's analysis of HF symmetry breaking; this state exhibits spatial and spin symmetry breaking. By varying a Thouless parameter, low degrees of symmetry breaking are introduced into ASDW states. After starting the dynamics from those states, higher degrees of symmetry breaking may subsequently emerge as dictated by the END equations without ad hoc interventions. Simulations starting from symmetry-conforming states preserve the symmetry features during dynamics, whereas simulations starting from symmetry-broken states display an upsurge of symmetry breaking once the reactants collide. Present simulations predict three types of reactions: (I) projectile scattering, (II) hydrogen substitution, and (III) water radiolysis into H + OH and 2H + O fragments. Remarkably, symmetry breaking considerably increases the extent of the target-to-projectile electron transfers (ETs) occurring during the above reactions. Then, with symmetry breaking, 1-ET differential and integral cross sections increase in value, whereas 0-ET differential cross sections and primary rainbow scattering angles decrease. More importantly, END properties calculated from symmetry-breaking simulations exhibit better agreement with the experimental data. Notably, END 1-ET integral cross sections with symmetry breaking compare better with their experimental counterparts than 1-ET integral cross sections from high-level close-coupling calculations; moreover, END validates an undetected rainbow scattering peak inferred from the experimental data. A discussion of our symmetry-breaking procedure in the context of Fukutome's analysis of HF symmetry breaking is also presented.