Number of unique cubes with one red cube in every $1*1*4$ segment

This is an offshoot of this question.

A $4*4*4$ cube must have exactly one red cube in every $1*1*4$ segment of the cube. By “segment” I mean any row, column or depth. There will thus be $16$ red cubes in total.

How many unique cubes are there which have this property?

A cube with this property is unique if it cannot be transformed into another cube with this property via rotations of the cube along one or more of its three central axes.

An example: Let’s take a smaller cube of size $2*2*2$. Such a cube has two solutions where every segment has exactly one red cube. But the solutions are not unique, as one could be turned into the other by a simple $90^\circ$ rotation of one face of the cube.

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