A topologically extendible mapping class that is not smoothly extendible

We give an example of a smooth characteristic embedding of a torus in $\s^2 \times \s^2 \# \s^1 \times \s^3$ such that there exists no diffeomorphism of the ambient $4$-manifold that induces the Dehn twist along a meridian of the torus, but there exists a homeomorphism of the ambient $4$-manifold, isotopic to identity, that induces the Dehn twist. As an application of our methods, we provide examples of two proper smooth embeddings of an annulus in $\s^2 \times \s^2 \# \s^1 \times \s^3 \setminus int(\D^4)$ which are topologically isotopic, but not smoothly isotopic (relative to boundary).