It is known ([1], [2]) that a construction of equivariant finiteness obstructions leads to a family $w_{\alpha }^H\left(X\right)$ of elements of the groups $K_0\left(\mathbb{Z}\left[{\pi }_0{\left(WH\left(X\right)\right)}_{\alpha }^{*}\right]\right)$. We prove that every family $w_{\alpha }^H$ of elements of the groups $K_0\left(\mathbb{Z}\left[{\pi }_0{\left(WH\left(X\right)\right)}_{\alpha }^{*}\right]\right)$ can be realized as the family of equivariant finiteness obstructions $w_{\alpha }^H\left(X\right)$ of an appropriate finitely dominated G-complex X. As an application of this result we show the natural equivalence of the geometric construction of equivariant finiteness obstruction ([5], [6]) and equivariant generalization of Wall’s obstruction...

For groups that satisfy the Isomorphism Conjecture in lower K-theory, we show that the cokernel of the forget-control K₀-groups is composed by the NK₀-groups of the finite subgroups. Using this information, we can calculate the exponent of each element in the cokernel in terms of the torsion of the group.