We investigate the time-resolved two-photon photoemission (time-resolved 2PPE, TR2PPE) spectrum of the image state on Cu(111) by nonequilibrium perturbation theory. We show that, in the 2PPE process, the scattering of photoexcited quasiparticles (electrons and holes) due to Coulomb interactions between electrons causes excitation of secondary quasiparticles in the vicinity of the Fermi level as well as electron transitions to the image state. We demonstrate quantitatively on the basis of numerical results how the effect of relaxation of secondary quasiparticles is manifested in the TR2PPE spectrum.