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 restriction map $mathrm{Cl}(X) to mathrm{Cl}(X_{Star(sigma)})$ in terms of the fans $Sigma$ and $Star(sigma)$?