Definition 2.6.1. A group $\Gamma$ is amenable if there exists a state $\mu$ on $l^{\infty}(\Gamma)$ which is invariant under the left translation action: for all $s\in \Gamma$ and $f\in l^{\infty}(\Gamma),~\mu(s.f)=\mu(f)$. (Here, the $s.f$ is the function $s.f(t)=f(s^{-1}t)$).
It is true that the class of amenable groups is closed under taking subgroups, extensions, quotients and inductive limits.
I have do some checking of the exercises above, but I am not sure whether my checking are correct ( I think the definition is a little abstract for me). So could someone help me to verify the exercises above?
This question is too broad and requires a lot of time answer it, so I suggest you to take a look at section 13 p.117 in Amenable locally compact groups. by Jean-Paul Pier