It is proved that, in any finite group G with nilpotent subgroups A and B and the condition A∩Bg⊴⟨A,Bg⟩ for any g in G, MinG(A,B) is a subgroup of F(G). This generalizes the author's theorem about intersections of Abelian subgroups in a finite group, since this holds, for example, for Hamiltonian subgroups A and B in G.