Inspired by this question, I wonder if anyone can provide an example of a finite CW complex X for which the order of the torsion subgroup of $H^{even} (X; \mathbb{Z}) = \bigoplus_{k=0}^\infty H^{2k} (X; \mathbb{Z})$ differs from the order of the torsion subgroup of $K^0 (X)$, where $K^0$ is complex topological K-theory. This is the same as asking for a non-zero differential in the Atiyah-Hirzebruch spectral sequence for some finite CW complex X, since this spectral sequence always collapses rationally.

Even better, is there an example in which X is a manifold? An orientable manifold?

Tom Goodwillie's answer to the question referenced above gave examples (real projective spaces) where the torsion subgroups are not isomorphic, but do have the same order.

It's interesting to note that the exponent of the images of these differentials is bounded by a universal constant, depending only on the starting page of the differential! This is a theorem of D. Arrlettaz (K-theory, 6: 347-361, 1992). You can even change the underlying spectrum (complex K-theory) without affecting the constant.