Terrence Tao provided an elementary proof of Gromov's theorem (https://terrytao.wordpress.com/2010/02/18/a-proof-of-gromovs-theorem/). I have been working my way through the proof and am stuck at the last two paragraphs which is pretty frustrating. In these paragraphs, Tao defines the group $G'':= \mathbb{Z} \ltimes_{e_m}N'$. As I understand it, the semi-direct product is defined by $(z, n)*(z', n') = (zz', e_m^zne_m^{-z}n')$, note that this may be wrong, I did not find this $\ltimes_g$ notation in other sources.
For more context, in the previous paragraphs, these objects are defined : $G'$ is a finite index subgroup of $G$, $\phi : G' \rightarrow \mathbb{Z}$ is a surjective homomorphism, $G = \langle e_1, ..., e_m \rangle$, $\ker(\phi) = \langle e_m^ke_je_m^{-k} | 1 \leq j \leq m-1, k \in \mathbb{Z} \rangle$. Finally, we define $N$ a nilpotent normal finite index ($M$) subgroup of $\ker(\phi)$ and $N' = \langle g^M | g \in \ker(\phi) \rangle \lhd N$.
My questions are the following, the proof says that $G''$ is a finite index subgroup of $G$. Not only do I not see how it is finite index... I don't even understand how it is a subgroup of $G$. Is there some embedding I do not know about ?
I hope my question is understandable and would be glad to get any advice.