It is well known that an extension of an amenable group by an amenable group is amenable. Is it possible to prove that by using only <a href="https://mathoverflow.net/questions/67524/a-paradoxical-decomposition-of-a-group">paradoxical decompositions</a>: if $G$ has a paradoxical decomposition and $N$ is a normal subgroup of $G$, then either $G/N$ or $N$ has paradoxical decomposition?