How to apply the Ford Fulkerson algorithm to the assignment problem
I suggest you start with a simpler problem: there are only two (possibly overlapping) subsets, A and B; and every group must have two objects: one from subset A, and one from B.
Once I figured out how to solve that, it was clear to me how to solve your original question.
Hint on how to solve that:

 Have one left vertex per object.  Have two right vertices per group, one to represent which object is chosen from subset A, and one to represent which object is chosen from subset B.  Now figure out how to fill in the edges...