Factor maps for automorphism groups via Cayley diagrams

We leverage a correspondence between group actions and edge-labelled graphs in two ways. First, we give a unified presentation of several folklore results connecting weak containment, local-global convergence, and continuous model theory. Second, we investigate the difference between $\operatorname{Aut}(\operatorname{Cay}(\Gamma))$-fiid combinatorics and $\Gamma$-fiid combinatorics for various marked groups $\Gamma$. It's straightforward to see that these differences vanish when $\operatorname{Cay}(\Gamma)$ admits an $\operatorname{Aut}(\operatorname{Cay}(\Gamma))$-fiid Cayley diagram. We extend this to show that the approximate combinatorics are the same when $\operatorname{Cay}(\Gamma)$ admits an approximate fiid Cayley diagram, and we give several examples and nonexamples of groups whose Cayley graphs admit (approximate) fiid Cayley diagrams. In particular, we show that trees admit approximate Cayley diagrams for any group whose Cayley graph is a tree; Cayley graphs of torsion free nilpotent groups do not admit fiid Cayley diagrams; and there are groups with isomorphic Cayley graphs so that only one them admits even an approximate Cayley diagram (in fact our construction answers a question of Weilacher).