I'm still trying to understand how to construct a factor group analysis for a given molecular point group and the resulting solid state group symmetry. For this I searched for some examples and found a good construction in a publication on $\ce{Na2SO4}$.
Sadly, the authors didn't give any information on the crystal structure and thus I had to test multiple versions. Their final group symmetry is given as $D_\mathrm{2h}$ which means it has to be a space group with the HM symbol of mmm. For $\ce{Na2SO4}$ this limits the selection to Cmcm ($\ce{Na2[CrO4]}$), Pbnm (olivine-structure) or Pbnn.
At this point I'm still unsure if the site-symmetry has some relation to the solid state group symmetry (any other than being a sub-group of it) or if it's related to the symmetry reduction of the molecular point-group by coordination.
An example of this is given here:
It's from K. Nakamoto, Infrared and Raman Spectra of Inorganic and Coordination Compounds and shows how the $T_\mathrm{d}$ symmetry of the free sulfate-anion is reduced by coordination to for example a metal. Now I always assumed that this is the first step you have to consider while constructing those factor group relationships.
So I took smaller and larger units from the crystal structure of all three given modifications of $\ce{Na2SO4}$ and made sure that they all have the sulfate sulfur in the center. Then I ran an automatic point group detection on them.
It seems like all three would return some symmetry elements and even using larger clusters resulted in the same symmetries (which you would probably expect from a periodic lattice).
The three site group (point groups) I could determine here were $C_\mathrm{s}$ and $C_\mathrm{2v}$. Besides that, comparing the non-trivial subgroups of $T_\mathrm{d}$ and $D_\mathrm{2h}$ would also offer $D_\mathrm{2}$ and $C_\mathrm{2}$.
When I compare this to the factor group analysis that Mabrouk et. Al. constructed in their Paper (DOI:0.1002/jrs.4374) it seems like they went for $D_\mathrm{2}$ only:
So either Mabrouk did a mistake while constructing them or used a crystal structure that is not in the ICSD or I'm wrong with my assumption that the site group symmetry is based on the symmetry reduction around the central ion of the molecular unit in question. Which is strange because other examples, even some starting from $T_\mathrm{d}$ arrive at the same conclusions as I when I use this method.
So if somebody has an idea where my mistake lies and how to correct it I'm open for any suggestions.
EDIT:
A minor editor here: I found another paper by R. L. Frost et. Al. about vanadates (DOI:10.1002/jrs.2906) and here the add more details to the site symmetry.
In vésigniéite,the two inequivalent Cu atoms and Ba atom occupy C2 sites whereas V, O1, O2, O3, O4 and H1 atoms occupy Cs sites.
I understand this explanation as occupying sites and then Cs is given in the following table as using the site symmetry as given by the space group. So in case of $\ce{Na2SO4}$ in Cmcm the Sulfur resides on 4c which has an m2m symmetry that equals $C_\mathrm{2v}$. The only real example I found would be for this Pbnn structure, where S resides on 4d which means it has a 2.. symmetry. I don't fully understand how the dots work in crystallography but if it means that they could be two as well then 222 = $D_\mathrm{2}$ which doesn't match my point group analysis though.
