  Extended [CII] emission on tens of kpc, also known as a [CII] halo, is being
currently reported around z$\sim$4-6 star-forming galaxies, especially thanks
to the statistics of the ALPINE survey. The [CII] emission is expected to trace
dense cold gas in the inner CGM of these galaxies. The origin of this emission
is still debated. In this paper, we present a post-processing model applied to
Illustris-TNG50 star-forming galaxies at $z\sim$4-6, and we compare our results
with the ALPINE observations. By incorporating C$^{+}$ abundances derived from
UV background and young stars as radiation sources, we generate mock
observations, from which we extract surface-brightness (SB) profiles. We found
that our model predicts similar [CII] emission values on galactic scales as the
observations, providing validation for our approach. However, we find that the
predicted [CII] emission in the inner circumgalactic medium (CGM) falls below
the observed values by a factor of $\sim$10. We discuss several model
limitations that may contribute to this discrepancy. We also find discrepancies
with observations when comparing SB profiles of low and high-SFR galaxies.
Unlike the observations, simulations exhibit no discernible difference in the
extended [CII] emission between the two subsamples. This discrepancy may
reflect shortcomings in feedback model of the simulation. Finally, our analysis
suggests that the extended [CII] emission is likely a result of both gas from
satellite galaxies and outflows from central galaxies, with satellites playing
a dominant role within 0.6$<$R/R$_{\rm vir}<$1. A firm estimate of the
importance of each contribution is beyond the scope of the current simulations.
