The conjugacy problem in automaton groups is not solvable

(Free abelian)-by-free, self-similar groups generated by finite self-similar sets of tree automorphisms and having unsolvable conjugacy problem are constructed. Along the way, finitely generated, orbit undecidable, free subgroups of GLd(Z), for d⩾6, and Aut(Fd), for d⩾5, are constructed as well.