Group theory - symmetry inequivalent sqaure matrices with PBC

I would like to obtain the symmetry inequivalent possible occupations of indistinguishable objects in a square lattice with periodic boundary conditions.

Examples:

• 1 object in a 4 by 4 square lattice has 16 equivalent positions -> 1 inequivalent position.
• 2 objects in a 3 by 3 square lattice has 2 inequivalent positions.

000    000
000    010
110    100

• 2 objects in a 4 by 4 square lattice has 5 inequivalent positions.

0000    0000    0000    0000    0000
0000    0000    0000    0000    0010
0000    0000    0100    0010    0000
1100    1010    1000    1000    1000


I can search for the inequivalent positions with a bruteforce approach by applying all possible symmetries to all possible initial configurations of 1 or 2, 3 objects and so on. Anyway, I was wondering if there is a way to do this with group theory in Mathematica.

I tried starting from this SO question but I didn't manage to get translations working.