Its not possible to make 6 circles on a crazy 4x4x4 typ III, i think.
The best is to make 4 circles around and top and down a pattern.
My avartar picture.
The reason is the geometry that ainner corner belong to a outer edge.
This was shown in this threadhttp://twistypuzzles.com/forum/viewtopic.php?f=15&t=14856&start=100
On crazy 3x3x3 plus ( 8 planets) Its possible to make 6 circles.
There are crazy 4x4x4 typ III puzzles on mefferts site. There are assembled in this way i think.