Let $(X,A)$ be a simply connected $CW$ pair such that $H_n(A)\cong H_n(X)$ for some $n$. I wonder if the isomorphism can be induced by inclusion $i:A\hookrightarrow X$ in this case.
Remark: Note that if this is always true, then by Hatcher's corollary 4.33, i:$A \hookrightarrow X$ is a homotopy equivalence, and thus by his corollary 0.20, $A$ is a deformation retract of $X$. This will generalize whitehead's theorem. So I feel like it isn't true. But I can't come up with an counterexample.