Let $X$ be a quasi-affine scheme; that is, the natural map $X\rightarrow \overline{X}:=Spec(\mathcal{O}_X)$ is an inclusion. Each scheme has a quasi-coherent sheaf of Kahler differentials $\Omega$, and the above open inclusion induces a $\mathcal{O}_X$-module map of global Kahler differentials

$$ \Gamma(\Omega_{\overline{X}})\rightarrow \Gamma(\Omega_{X})$$

Is this map always an isomorphism?

Let $X$ be a quasi-affine scheme; that is, the natural map $$X\rightarrow \overline{X}:=Spec(\Gamma(X,\mathcal{O}_X))$$ is an inclusion. Each scheme has a quasi-coherent sheaf of Kahler differentials $\Omega$, and the above open inclusion induces a $\Gamma(X,\mathcal{O}_X)$-module map of global Kahler differentials

$$ \Gamma(\Omega_{\overline{X}})\rightarrow \Gamma(\Omega_{X})$$

Is this map always an isomorphism?

Edit: Changed $\mathcal{O}_X$ to $\Gamma(X,\mathcal{O}_X)$.