How should I correct Syzygy stats?

I want to calculate some stats using the .json files from the site https://syzygy-tables.info/.

I want to calculate for a class of endgames, so using the percentages from the individual endgames is not sufficient, because the total number of positions in each endgame is variable. I would therefore like to use the actual numbers of positions.

The actual numbers of positions are purportedly given in the .json files. My problem is that these are generally incorrect. 

The error appears to be connected with the fact that to conserve space in the Syzygy tables, for most positions only one of a collection of symmetrical positions is held. The reporting routine appears to use the reduced set without correcting for this. 

Unfortunately the correction to be applied appears to vary by endgame, so I'm hoping that someone who understands how the Syzygy tables are produced can tell me how this works.

To illustrate I show below two cases that are easy to calculate on the back of an envelope, together with Syzygy reported figures and figures from Wilhelm/Nalimov (which do make the correction).

Number of won positions in KNNK

Back of envelope: 736

Syzygy: 184

Wilhelm: 736

Number of drawn positions in KRK

Back of envelope: 22,228

Syzygy: 2,796

Wilhelm: 22,244

Notice that the Syzygy figure is exactly one quarter of the correct (BOE) figure in the first case and very close to one eighth of the correct figure in the second.

The discrepancy between the BOE and Wilhelm figures in the second case is the inclusion of 16 positions in the Wilhelm figure that are illegal with any FRC starting position and is to be expected. There are no illegal won positions in the first case.

The Wilhelm figure in the second case can be reproduced from the Syzygy figure by multiplying by 8 (the order of the symmetry group of the board) and subtracting 124 (the number of draws with bilateral symmetry). The same can be done in the first case by multiplying by 8 and dividing by 2!=2 (the order of the group of permutations of the knights?). If this is the procedure for pawnless endgames the question remains should the symmetric draws also be adjusted by n! for each repeated piece?

One would expect a different procedure for endgames with pawns because the board symmetries are reduced.

Wins in KPK

Back of envelope: In progress

Syzygy: 111, 282 

Wilhelm: 222,564 

Note that the Wilhelm figure in this case is exactly twice the Syzygy figure - 488 symmetrical wins not subtracted. I should add the lack of subtraction here is not surprising because the table can only use symmetry between the flanks, whereas symmetrical positions are necessarily reflections in diagonals since each side has only one king.

Can anybody clarify?