Amenable group which is inverse limit of non-amenable groups

Question

This question is the dual of the following question Example of non-amenable group which is the inverse limit of amenable groups ; in the answer it is claimed that an inverse limit of amenable groups can be non-amenable. Let us suppose we have an inverse system of discrete groups $G_i$ which is semistable, i.e. all the maps are surjective. Is it possibile that $\underset{\leftarrow}{\lim}G_i$ is amenable but infinite of the $G_i$ are non-amenable?

Answer 1

Let us denote with $G$ the inverse limit; thanks to the universal property and since all the maps in the system are surjective, we have surjective and continuous projections $\pi_i:G\rightarrow G_i$. So for each $i$ we have $G_i\simeq G/ Ker(\pi_i)$. So each $G_i$ is a quotient of an amenable group by a normal closed subgroup, hence it is amenable.