Given a countable amenable group G one can ask which are the real numbers that can be realized as the topological entropy of a subshift of finite type (SFT). A famous result by Hochman and Meyerovitch completely characterizes these numbers for Z2. I will show that the same characterization is valid for any amenable group with decidable word problem which admits an action of Z2 which is free and bounded. Using this result we can give a full characterization of the entropies of SFTs for polycyclic groups. Furthermore, the same result holds for any countable group with decidable word problem which contains the direct product of any pair of infinite, finitely generated and amenable groups. In particular, it holds for many branch groups such as the Grigorchuk group.