  Molecular hydrogen allows cooling in primordial gas, facilitating its
collapse into Population III stars within primordial halos. Lyman-Werner (LW)
radiation from these stars can escape the halo and delay further star formation
by destroying H$_2$ in other halos. As cosmological simulations show that
increasing the background LW field strength increases the average halo mass
required for star formation, we perform follow-up simulations of selected halos
to investigate the knock-on effects this has on the Population III IMF. We
follow 5 halos for each of the $J_{21}$ = 0, 0.01 and 0.1 LW field strengths,
resolving the pre-stellar core density of $10^{-6}$ g cm$^{-3}$ (10$^{18}$
cm$^{-3}$) before inserting sink particles and following the fragmentation
behaviour for hundreds of years further. We find that the mass accreted onto
sinks by the end of the simulations is proportional to the mass within the
$\sim 10^{-2}$ pc molecular core, which is not correlated to the initial mass
of the halo. As such, the IMFs for masses above the brown dwarf limit show
little dependence on the LW strength, although they do show variance in the
number of low-mass clumps formed. As the range of background LW field strengths
tested here covers the most likely values from literature, we conclude that the
IMF for so-called Pop III.2 stars is not significantly different from the
initial population of Pop III.1 stars. The primordial IMF therefore likely
remains unchanged until the formation of the next generation of Population II
stars.
