  We study how a cosmological bounce with a Type IV singularity at the bouncing
point, can be generated by a classical vacuum $F(G)$ gravity. We focus our
investigation on the behavior of the vacuum $F(G)$ theory near the Type IV
singular bouncing point and also we address the stability of the resulting
solution, by treating the equations of motion as a dynamical system. In
addition, we investigate how the scalar perturbations of the background metric
evolve, emphasizing to cosmological times near the Type IV singular bouncing
point. Finally, we also investigate which mimetic vacuum $F(G)$ gravity can
describe the singular bounce cosmology.
