We have theoretically studied the population transfer in a five-level ${\mathrm{Li}}_{2}$ dimer by $(2+2)$-photon stimulated hyper-Raman nonadiabatic passage (STIHRNAP) from the initial ground level ${v}_{g}=0$, ${J}_{g}=0$ to the final rovibrational levels ${v}_{f}=1$ (2), ${J}_{f}=0,2$, of the ground electronic state $X\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}_{g}^{+}$ via the resonant intermediate levels ${v}_{i}=1$, ${J}_{i}=0,2$, of the excited electronic state $2\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}_{g}^{+}$. Linearly chirped pump and Stokes laser pulses with different chirp rates and without any initial detuning are applied simultaneously. Both the pulses are taken to have the same temporal shape, pulse width, and linear parallel polarizations. The density matrix method has been used to compute the populations of the levels. We have investigated in detail the population transfer for laser wavelengths (at time $t=0$) in the range of $992\char21{}1028\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ and (peak) intensities in the range of $1.5\ifmmode\times\else\texttimes\fi{}{10}^{10}\char21{}3.0\ifmmode\times\else\texttimes\fi{}{10}^{11}\phantom{\rule{0.3em}{0ex}}\mathrm{W}∕{\mathrm{cm}}^{2}$. The required pulse widths are of the order of $35\char21{}275\phantom{\rule{0.3em}{0ex}}\mathrm{ps}$ for maximum population inversion. We have controlled rotational branching in population transfer to the final rotational levels ${J}_{f}=0$ ($Q$ branch) and ${J}_{f}=2$ ($S$ branch) of the fundamental $({v}_{f}=1)$ and first overtone $({v}_{f}=2)$ transitions by judicious choice of laser parameters. We have applied a $(2+2)$-photon STIHRNAP process to a model multilevel molecular system $({\mathrm{Li}}_{2})$ and achieved almost complete population transfer from the initial ground to the final target rovibrational levels with chirped ir laser pulses.