(a) Suppose $\mathrm{X}=\operatorname{Spec} \mathrm{k}[\mathrm{t}]$. Let $\mathscr{F}$ be the skyscraper sheaf supported at the origin $[(t)]$, with group $k(t)$ and the usual $k[t]$-module structure. Show that this is an $\sigma_{\mathrm{x}}$-module that is not a quasicoherent sheaf. (More generally, if X is an integral scheme, and $p \in X$ is not the generic point, we could take the skyscraper sheaf at $p$ with group the function field of $X$. Except in silly circumstances, this sheaf won't be quasicoherent.) See Exercises 9.1.G and 6.2.H for more (pathological) examples of $\mathscr{U}_{\mathrm{x}}$-modules that are not quasicoherent.
(b) Suppose $\mathrm{X}=\operatorname{Spec} \mathrm{k}[\mathrm{t}]$. Let $\mathscr{F}$ be the skyscraper sheaf supported at the generic point $[(0)]$, with group $\mathrm{k}(\mathrm{t})$. Give this the structure of an $\mathscr{U}_{\mathrm{X}}$-module. Show that this is a quasicoherent sheaf. Describe the restriction maps in the distinguished topology of X. (Remark: Your argument will apply more generally, for example when $X$ is an integral scheme with generic point $\eta$, and $\mathscr{F}$ is the skyscraper sheaf $\mathrm{i}_{\mathrm{\eta}, \mathrm{a}} \mathrm{K}(\mathrm{X})$.)