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.