3 added 164 characters in body

Concerning the analogous question for cohomology: There is a homeomorphism from the Klein bottle to itself that induces the identity map in integral cohomology but not in mod $2$ cohomology.

This example is Spanier-Whitehead dual to the $S^2\vee \mathbb RP^2$ variant of Sam's example.

For an example of a map inducing the identity in both integral homology and in integral cohomology, but not in mod $2$ (co)homology, you need to use a homology group that is not finitely generated.

In Sam's example (map from $\mathbb RP^2\vee \Sigma \mathbb RP^2$ to itself) the map is zero on both integral homology and integral cohomology.

2 added 60 characters in body

Concerning the analogous question for cohomology: There is a homeomorphism from the Klein bottle to itself that induces the identity map in integral cohomology but not in mod $2$ cohomology.

This example is Spanier-Whitehead dual to the $S^2\vee \mathbb RP^2$ variant of Sam's example.

For an example of a map inducing the identity in both integral homology and in integral cohomology, but not in mod $2$ (co)homology?co)homology, you need to use a homology group that is not finitely generated.

1

Concerning the analogous question for cohomology: There is a homeomorphism from the Klein bottle to itself that induces the identity map in integral cohomology but not in mod $2$ cohomology.

This example is Spanier-Whitehead dual to the $S^2\vee \mathbb RP^2$ variant of Sam's example.

Now how about an example of a map inducing the identity in integral homology and in integral cohomology, but not in mod $2$ (co)homology?