Let $X$ be a (say, complex) toric variety acted upon by a torus $T$ and defined by a fan $\Sigma$ in the cocharacter lattice $N=\mathrm{Hom}(\mathbb{C}^\times, T)$, and let $M$ be the character lattice. For any cone $\sigma \in \Sigma$ put $M(\sigma) = \sigma^\perp \cap M$, $N(\sigma) = \mathrm{Hom}(M(\sigma), \mathbb{C}^\times)$. There is a natural projection $N \to N(\sigma)$. Then the closure of the orbit corresponding to $\sigma$ has the structure of a toric variety with respect to the quotient torus with the cocharacter lattice $N/N(\sigma)$ and given by the fan $Star(\sigma)$ consisting of the images in $N(\sigma)$ of the cones of $\Sigma$ containing $\sigma$. Note that the closed embedding $X_{Star(\sigma)} \to X$  is generally not a toric morphism, since the dense toric orbit of $X_{Star(\sigma)}$ does not intersect the dense toric orbit of $X$.

My question is: is there a way to describe the map $\mathrm{Cl}(X) \to \mathrm{Cl}(X_{Star(\sigma)})$ in terms of the fans $\Sigma$ and $Star(\sigma)$?