Colorings breaking prescribed families of graph automorphisms

We prove that for every continuous action of a compact group on a discrete set, each finite-color template admits a coloring from arbitrary lists of the same size whose stabilizer is contained in a conjugate of the template stabilizer. Consequently, the ordinary and list parameters coincide whenever the ordinary parameter is finite. In particular, we answer the question of Ferrara, Flesch and Gethner negatively for finite graphs, and establish the finite case of the conjecture of Kwaśny and Stawiski whenever a distinguishing edge coloring exists. We also show that every closed locally compact permutation group with infinite motion on a countably infinite set admits continuum many inequivalent distinguishing colorings from arbitrary 2-element lists. Using compact kernels and finite bases, we extend ordinary and list equality to countable graphs with countable automorphism groups and prove that one additional list color suffices for connected locally finite graphs. Using this multiplicity and a construction fixing all vertices of infinite degree, we prove that \(D'_\ell(G)\le 2\) for every countable connected graph with infinite edge motion, settling the list infinite edge-motion conjecture of Kwaśny and Stawiski.