$\mathbf {The \ Problem \ is}:$ Let, $(M,g)$ be a homogeneous Riemannian manifold and $\Gamma$ be a discrete subgroup of Isom$(M,g):=G.$ Let, the normaliser of $\Gamma = N_G(\Gamma):=N$ acts transitively on $M.$ Then show $(\bar{M}:=M/{\Gamma},\bar{g})$ is also a homogeneous space.
Give an example of $(M,g)$ where $N$ acts transitively on $M.$
$\mathbf {My \ approach}:$ I can't find where to exactly use the fact that $N$ acts transitively on $M.$
Since, the quotient map $q:M\to M/{\Gamma}$ is a local isometry (because $\Gamma$ acts freely and properly discontinuously on $M$) and $M$ is homogeneous then it turns out $M/{\Gamma}$ is also homogeneous.
I can't find where I am doing wrong ?
A hint for both the problem and the example is very much needed. Thanks in advance.