A topologically extendible mapping class that is not smoothly extendible

We give an example of a smooth characteristic embedding of a torus in ${\text{s}}^2\times {\text{s}}^2\mathrm{\#}{\text{s}}^1\times {\text{s}}^3$ 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 ${\text{s}}^2\times {\text{s}}^2\mathrm{\#}{\text{s}}^1\times {\text{s}}^3\setminus int({\text{D}}^4)$ which are topologically isotopic, but not smoothly isotopic (relative to boundary).