Let $F$ be a free group on $d$ generators.

Denote by $F_{k}$ the $k$-th term in $F$'s derived series. Put $G = F/F_k$. What is the normal subgroup growth of $G$? Explicitly, for each natural number $n$, what is the number of normal subgroups of index not greater than $n$ in $G$? An asymptotic result will also be interesting of course.

For example, the case $k=2$ is covered in the book about subgroup growth.

More generally, Let $C$ be a formation of finite groups ($C$ is closed for quotients and subdirect products). Let $G$ be the quotient of $F$ by the intersection of all normal subgroups $N \unlhd F$ for which $F/N$ is a $C$-group. Assume that $G$ is not trivial ($F$ is not residually $C$). The question is the same as before: What is the normal subgroup growth of $G$?