Consider six countable Abelian groups and six group homomorphims as in the following diagram

G → H → I ↑ ↓ L ← K ← J

Assume that the resulting sequence is exact at all six entries.

Question: Is there a (second countable) locally compact Hausdorff space X with a closed subspace A, such that the resulting six-term sequence in K-theory

K^{0}(X,A) → K^{0}(X) → K^{0}(A) ↑ ↓ K^{1}(A) ← K^{1}(X) ← K^{1}(X,A)

is isomorphic to the above one?

An answer in the finitely generated case would also be interesting.