dimension=4;

none=0;
black=1;
white=2;

permutations=Permutations[{black,black,black,black,black,black,black,black,white,white,white,white,white,white,white,white}];

Length[permutations]

group=
{{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16},{1,2,5,6,9,10,13,14,3,4,7,8,11,12,15,16},{1,2,9,10,3,4,11,12,5,6,13,14,7,8,15,16},{1,3,2,4,9,11,10,12,5,7,6,8,13,15,14,16},{1,3,5,7,2,4,6,8,9,11,13,15,10,12,14,16},{1,3,9,11,5,7,13,15,2,4,10,12,6,8,14,16},{1,5,2,6,3,7,4,8,9,13,10,14,11,15,12,16},{1,5,3,7,9,13,11,15,2,6,4,8,10,14,12,16},{1,5,9,13,2,6,10,14,3,7,11,15,4,8,12,16},{1,9,2,10,5,13,6,14,3,11,4,12,7,15,8,16},{1,9,3,11,2,10,4,12,5,13,7,15,6,14,8,16},{1,9,5,13,3,11,7,15,2,10,6,14,4,12,8,16},{2,1,4,3,10,9,12,11,6,5,8,7,14,13,16,15},{2,1,6,5,4,3,8,7,10,9,14,13,12,11,16,15},{2,1,10,9,6,5,14,13,4,3,12,11,8,7,16,15},{2,4,1,3,6,8,5,7,10,12,9,11,14,16,13,15},{2,4,6,8,10,12,14,16,1,3,5,7,9,11,13,15},{2,4,10,12,1,3,9,11,6,8,14,16,5,7,13,15},{2,6,1,5,10,14,9,13,4,8,3,7,12,16,11,15},{2,6,4,8,1,5,3,7,10,14,12,16,9,13,11,15},{2,6,10,14,4,8,12,16,1,5,9,13,3,7,11,15},{2,10,1,9,4,12,3,11,6,14,5,13,8,16,7,15},{2,10,4,12,6,14,8,16,1,9,3,11,5,13,7,15},{2,10,6,14,1,9,5,13,4,12,8,16,3,11,7,15},{3,1,4,2,7,5,8,6,11,9,12,10,15,13,16,14},{3,1,7,5,11,9,15,13,4,2,8,6,12,10,16,14},{3,1,11,9,4,2,12,10,7,5,15,13,8,6,16,14},{3,4,1,2,11,12,9,10,7,8,5,6,15,16,13,14},{3,4,7,8,1,2,5,6,11,12,15,16,9,10,13,14},{3,4,11,12,7,8,15,16,1,2,9,10,5,6,13,14},{3,7,1,5,4,8,2,6,11,15,9,13,12,16,10,14},{3,7,4,8,11,15,12,16,1,5,2,6,9,13,10,14},{3,7,11,15,1,5,9,13,4,8,12,16,2,6,10,14},{3,11,1,9,7,15,5,13,4,12,2,10,8,16,6,14},{3,11,4,12,1,9,2,10,7,15,8,16,5,13,6,14},{3,11,7,15,4,12,8,16,1,9,5,13,2,10,6,14},{4,2,3,1,12,10,11,9,8,6,7,5,16,14,15,13},{4,2,8,6,3,1,7,5,12,10,16,14,11,9,15,13},{4,2,12,10,8,6,16,14,3,1,11,9,7,5,15,13},{4,3,2,1,8,7,6,5,12,11,10,9,16,15,14,13},{4,3,8,7,12,11,16,15,2,1,6,5,10,9,14,13},{4,3,12,11,2,1,10,9,8,7,16,15,6,5,14,13},{4,8,2,6,12,16,10,14,3,7,1,5,11,15,9,13},{4,8,3,7,2,6,1,5,12,16,11,15,10,14,9,13},{4,8,12,16,3,7,11,15,2,6,10,14,1,5,9,13},{4,12,2,10,3,11,1,9,8,16,6,14,7,15,5,13},{4,12,3,11,8,16,7,15,2,10,1,9,6,14,5,13},{4,12,8,16,2,10,6,14,3,11,7,15,1,9,5,13},{5,1,6,2,13,9,14,10,7,3,8,4,15,11,16,12},{5,1,7,3,6,2,8,4,13,9,15,11,14,10,16,12},{5,1,13,9,7,3,15,11,6,2,14,10,8,4,16,12},{5,6,1,2,7,8,3,4,13,14,9,10,15,16,11,12},{5,6,7,8,13,14,15,16,1,2,3,4,9,10,11,12},{5,6,13,14,1,2,9,10,7,8,15,16,3,4,11,12},{5,7,1,3,13,15,9,11,6,8,2,4,14,16,10,12},{5,7,6,8,1,3,2,4,13,15,14,16,9,11,10,12},{5,7,13,15,6,8,14,16,1,3,9,11,2,4,10,12},{5,13,1,9,6,14,2,10,7,15,3,11,8,16,4,12},{5,13,6,14,7,15,8,16,1,9,2,10,3,11,4,12},{5,13,7,15,1,9,3,11,6,14,8,16,2,10,4,12},{6,2,5,1,8,4,7,3,14,10,13,9,16,12,15,11},{6,2,8,4,14,10,16,12,5,1,7,3,13,9,15,11},{6,2,14,10,5,1,13,9,8,4,16,12,7,3,15,11},{6,5,2,1,14,13,10,9,8,7,4,3,16,15,12,11},{6,5,8,7,2,1,4,3,14,13,16,15,10,9,12,11},{6,5,14,13,8,7,16,15,2,1,10,9,4,3,12,11},{6,8,2,4,5,7,1,3,14,16,10,12,13,15,9,11},{6,8,5,7,14,16,13,15,2,4,1,3,10,12,9,11},{6,8,14,16,2,4,10,12,5,7,13,15,1,3,9,11},{6,14,2,10,8,16,4,12,5,13,1,9,7,15,3,11},{6,14,5,13,2,10,1,9,8,16,7,15,4,12,3,11},{6,14,8,16,5,13,7,15,2,10,4,12,1,9,3,11},{7,3,5,1,15,11,13,9,8,4,6,2,16,12,14,10},{7,3,8,4,5,1,6,2,15,11,16,12,13,9,14,10},{7,3,15,11,8,4,16,12,5,1,13,9,6,2,14,10},{7,5,3,1,8,6,4,2,15,13,11,9,16,14,12,10},{7,5,8,6,15,13,16,14,3,1,4,2,11,9,12,10},{7,5,15,13,3,1,11,9,8,6,16,14,4,2,12,10},{7,8,3,4,15,16,11,12,5,6,1,2,13,14,9,10},{7,8,5,6,3,4,1,2,15,16,13,14,11,12,9,10},{7,8,15,16,5,6,13,14,3,4,11,12,1,2,9,10},{7,15,3,11,5,13,1,9,8,16,4,12,6,14,2,10},{7,15,5,13,8,16,6,14,3,11,1,9,4,12,2,10},{7,15,8,16,3,11,4,12,5,13,6,14,1,9,2,10},{8,4,6,2,7,3,5,1,16,12,14,10,15,11,13,9},{8,4,7,3,16,12,15,11,6,2,5,1,14,10,13,9},{8,4,16,12,6,2,14,10,7,3,15,11,5,1,13,9},{8,6,4,2,16,14,12,10,7,5,3,1,15,13,11,9},{8,6,7,5,4,2,3,1,16,14,15,13,12,10,11,9},{8,6,16,14,7,5,15,13,4,2,12,10,3,1,11,9},{8,7,4,3,6,5,2,1,16,15,12,11,14,13,10,9},{8,7,6,5,16,15,14,13,4,3,2,1,12,11,10,9},{8,7,16,15,4,3,12,11,6,5,14,13,2,1,10,9},{8,16,4,12,7,15,3,11,6,14,2,10,5,13,1,9},{8,16,6,14,4,12,2,10,7,15,5,13,3,11,1,9},{8,16,7,15,6,14,5,13,4,12,3,11,2,10,1,9},{9,1,10,2,11,3,12,4,13,5,14,6,15,7,16,8},{9,1,11,3,13,5,15,7,10,2,12,4,14,6,16,8},{9,1,13,5,10,2,14,6,11,3,15,7,12,4,16,8},{9,10,1,2,13,14,5,6,11,12,3,4,15,16,7,8},{9,10,11,12,1,2,3,4,13,14,15,16,5,6,7,8},{9,10,13,14,11,12,15,16,1,2,5,6,3,4,7,8},{9,11,1,3,10,12,2,4,13,15,5,7,14,16,6,8},{9,11,10,12,13,15,14,16,1,3,2,4,5,7,6,8},{9,11,13,15,1,3,5,7,10,12,14,16,2,4,6,8},{9,13,1,5,11,15,3,7,10,14,2,6,12,16,4,8},{9,13,10,14,1,5,2,6,11,15,12,16,3,7,4,8},{9,13,11,15,10,14,12,16,1,5,3,7,2,6,4,8},{10,2,9,1,14,6,13,5,12,4,11,3,16,8,15,7},{10,2,12,4,9,1,11,3,14,6,16,8,13,5,15,7},{10,2,14,6,12,4,16,8,9,1,13,5,11,3,15,7},{10,9,2,1,12,11,4,3,14,13,6,5,16,15,8,7},{10,9,12,11,14,13,16,15,2,1,4,3,6,5,8,7},{10,9,14,13,2,1,6,5,12,11,16,15,4,3,8,7},{10,12,2,4,14,16,6,8,9,11,1,3,13,15,5,7},{10,12,9,11,2,4,1,3,14,16,13,15,6,8,5,7},{10,12,14,16,9,11,13,15,2,4,6,8,1,3,5,7},{10,14,2,6,9,13,1,5,12,16,4,8,11,15,3,7},{10,14,9,13,12,16,11,15,2,6,1,5,4,8,3,7},{10,14,12,16,2,6,4,8,9,13,11,15,1,5,3,7},{11,3,9,1,12,4,10,2,15,7,13,5,16,8,14,6},{11,3,12,4,15,7,16,8,9,1,10,2,13,5,14,6},{11,3,15,7,9,1,13,5,12,4,16,8,10,2,14,6},{11,9,3,1,15,13,7,5,12,10,4,2,16,14,8,6},{11,9,12,10,3,1,4,2,15,13,16,14,7,5,8,6},{11,9,15,13,12,10,16,14,3,1,7,5,4,2,8,6},{11,12,3,4,9,10,1,2,15,16,7,8,13,14,5,6},{11,12,9,10,15,16,13,14,3,4,1,2,7,8,5,6},{11,12,15,16,3,4,7,8,9,10,13,14,1,2,5,6},{11,15,3,7,12,16,4,8,9,13,1,5,10,14,2,6},{11,15,9,13,3,7,1,5,12,16,10,14,4,8,2,6},{11,15,12,16,9,13,10,14,3,7,4,8,1,5,2,6},{12,4,10,2,16,8,14,6,11,3,9,1,15,7,13,5},{12,4,11,3,10,2,9,1,16,8,15,7,14,6,13,5},{12,4,16,8,11,3,15,7,10,2,14,6,9,1,13,5},{12,10,4,2,11,9,3,1,16,14,8,6,15,13,7,5},{12,10,11,9,16,14,15,13,4,2,3,1,8,6,7,5},{12,10,16,14,4,2,8,6,11,9,15,13,3,1,7,5},{12,11,4,3,16,15,8,7,10,9,2,1,14,13,6,5},{12,11,10,9,4,3,2,1,16,15,14,13,8,7,6,5},{12,11,16,15,10,9,14,13,4,3,8,7,2,1,6,5},{12,16,4,8,10,14,2,6,11,15,3,7,9,13,1,5},{12,16,10,14,11,15,9,13,4,8,2,6,3,7,1,5},{12,16,11,15,4,8,3,7,10,14,9,13,2,6,1,5},{13,5,9,1,15,7,11,3,14,6,10,2,16,8,12,4},{13,5,14,6,9,1,10,2,15,7,16,8,11,3,12,4},{13,5,15,7,14,6,16,8,9,1,11,3,10,2,12,4},{13,9,5,1,14,10,6,2,15,11,7,3,16,12,8,4},{13,9,14,10,15,11,16,12,5,1,6,2,7,3,8,4},{13,9,15,11,5,1,7,3,14,10,16,12,6,2,8,4},{13,14,5,6,15,16,7,8,9,10,1,2,11,12,3,4},{13,14,9,10,5,6,1,2,15,16,11,12,7,8,3,4},{13,14,15,16,9,10,11,12,5,6,7,8,1,2,3,4},{13,15,5,7,9,11,1,3,14,16,6,8,10,12,2,4},{13,15,9,11,14,16,10,12,5,7,1,3,6,8,2,4},{13,15,14,16,5,7,6,8,9,11,10,12,1,3,2,4},{14,6,10,2,13,5,9,1,16,8,12,4,15,7,11,3},{14,6,13,5,16,8,15,7,10,2,9,1,12,4,11,3},{14,6,16,8,10,2,12,4,13,5,15,7,9,1,11,3},{14,10,6,2,16,12,8,4,13,9,5,1,15,11,7,3},{14,10,13,9,6,2,5,1,16,12,15,11,8,4,7,3},{14,10,16,12,13,9,15,11,6,2,8,4,5,1,7,3},{14,13,6,5,10,9,2,1,16,15,8,7,12,11,4,3},{14,13,10,9,16,15,12,11,6,5,2,1,8,7,4,3},{14,13,16,15,6,5,8,7,10,9,12,11,2,1,4,3},{14,16,6,8,13,15,5,7,10,12,2,4,9,11,1,3},{14,16,10,12,6,8,2,4,13,15,9,11,5,7,1,3},{14,16,13,15,10,12,9,11,6,8,5,7,2,4,1,3},{15,7,11,3,16,8,12,4,13,5,9,1,14,6,10,2},{15,7,13,5,11,3,9,1,16,8,14,6,12,4,10,2},{15,7,16,8,13,5,14,6,11,3,12,4,9,1,10,2},{15,11,7,3,13,9,5,1,16,12,8,4,14,10,6,2},{15,11,13,9,16,12,14,10,7,3,5,1,8,4,6,2},{15,11,16,12,7,3,8,4,13,9,14,10,5,1,6,2},{15,13,7,5,16,14,8,6,11,9,3,1,12,10,4,2},{15,13,11,9,7,5,3,1,16,14,12,10,8,6,4,2},{15,13,16,14,11,9,12,10,7,5,8,6,3,1,4,2},{15,16,7,8,11,12,3,4,13,14,5,6,9,10,1,2},{15,16,11,12,13,14,9,10,7,8,3,4,5,6,1,2},{15,16,13,14,7,8,5,6,11,12,9,10,3,4,1,2},{16,8,12,4,14,6,10,2,15,7,11,3,13,5,9,1},{16,8,14,6,15,7,13,5,12,4,10,2,11,3,9,1},{16,8,15,7,12,4,11,3,14,6,13,5,10,2,9,1},{16,12,8,4,15,11,7,3,14,10,6,2,13,9,5,1},{16,12,14,10,8,4,6,2,15,11,13,9,7,3,5,1},{16,12,15,11,14,10,13,9,8,4,7,3,6,2,5,1},{16,14,8,6,12,10,4,2,15,13,7,5,11,9,3,1},{16,14,12,10,15,13,11,9,8,6,4,2,7,5,3,1},{16,14,15,13,8,6,7,5,12,10,11,9,4,2,3,1},{16,15,8,7,14,13,6,5,12,11,4,3,10,9,2,1},{16,15,12,11,8,7,4,3,14,13,10,9,6,5,2,1},{16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1}};

Length[group]

list=permutations;

ret={};

loop=0;

While[list≠{},
obj=First[list];
AppendTo[ret,obj];
list=Rest[list];
pattern=Alternatives@@Union[obj[[#]]&/@group];
list=DeleteCases[list,pattern];
loop++;
]

initial=ret

Length[initial]

patternList=initial;

initialBoard[n_,type_]:=Block[{ret},
ret=ConstantArray[none,ConstantArray[n,dimension]];
MapThread[(ret[[Sequence@@(ConstantArray[n/2,dimension]+#1)]]=#2)&,{Tuples[{0,1},dimension],patternList[[type]]}];
Return[ret];
]

initialBoard[4,1]

initialBoard[4,96]

initialBoard[4,97]

Manipulate[MatrixForm[initialBoard[4,type]],{type,1,Length[patternList],1}]