A Cayley diagram is a labeling of a graph $G$ that encodes an action of a group which induces $G$. For instance, a $d$-edge coloring of a $d$-regular tree is a Cayley diagram for the group $(\mathbb{Z}/2\mathbb{Z})^{*d}$. In this talk, we will investigate when a Cayley graph $G=(\Gamma, E)$ admits an $\operatorname{Aut}(G)$-f.i.i.d. Cayley diagram and show that $\Gamma$-f.i.i.d. solutions to local labeling problems for such graphs lift to $\operatorname{Aut}(G)$-f.i.i.d. solutions.