The cohomological restriction map and FP-infinity groups.

Let G be a group and let H be a subgroup of G of finite index. As a result of Quillen's Theorem we know that if G is finite, then the restriction map from the cohomology ring of G to that of H has a finitely generated kernel. Following Bartholdi, we ask whether this is true for an arbitrary group G. We will show that this is true in case the group G is of type FP∞ and has virtual finite cohomological dimension, and we will give two counterexamples for the general case, one in which G is not finitely generated, and one in which the group G is an FP∞ group.