Let $G$ be a group and let $L$ be a normal subgroup of $G$. Moreover, I have normal finite index subgroups $N_{i}$ of $G$ for $i\in \{1,\cdots,n\}$ such that the $n$-fold commutator $$[N_{1},\cdots,N_{n}]\subseteq L.$$
I want to show that $G/L$ is virtually a nilpotent group. For that, my idea was to work with the group $$G/[N_{1},\cdots,N_{n}],$$ and then to consider the natural surjective map $$G/[N_{1},\cdots,N_{n}] \mapsto G/L.$$ Nevertheless, I am stuck in trying to prove that $G/[N_{1},\cdots,N_{n}]$ is virtually nilpotent. Can someone help me, please?
The intersection $N = N_1 \cap N_2 \cap \cdots \cap N_n$ of the $N_i$ has finite index in $G$, and $[N,N,\ldots,N] \le L$, so $[NL,NL,\cdots,NL] \le L$ and hence $NL/L$ is nilpotent (of class at most $n-1$) and has finite index in $G/L$.