My question is rather general and the reason for this is that I am primarily interested in finding sources where I can read more about this type of problems. So here goes:

To begin with, I want to find out more about methods to determine the number of conjugacy classes of subgroups of a given order in an ambient group $G$. A specific question I have in this vein is this:

Suppose $G$ is a finite 2-group and $N \triangleleft G$ is a normal subgroup of $G$ such that $G/N \cong N$. Now assume that $N$ has $f(i)$ conjugacy classes of subgroups of order $2^i$, $i \in \{0,1, \dots , \log_2|N|\}$. Is it true that $G$ has (at most) $f(i)f(j)$ conjugacy classes of subgroups of order $2^{i+j}$?

If the answer is "no" in general, is it possible to make further assumptions on either $G$ or $N$ in order to obtain an affirmative answer?

My question is rather general and the reason for this is that I am primarily interested in finding sources where I can read more about this type of problems. So here goes:

To begin with, I want to find out more about methods to determine the number of conjugacy classes of subgroups of a given order in an ambient group $G$. A specific question I have in this vein is this:

Suppose $G$ is a finite 2-group and $N \triangleleft G$ is a normal subgroup of $G$ such that $G/N \cong N$. Now assume that $N$ has $f(i)$ conjugacy classes of subgroups of order $2^i$, $i \in \{0,1, \dots , \log_2|N|\}$. Is it true that $G$ has (at most) $f(i)f(j)$ conjugacy classes of subgroups of order $2^{i+j}$?

If the answer is "no" in general, is it possible to make further assumptions on either $G$ or $N$ in order to obtain an affirmative answer?

P.S. You may assume that $N=Z(G)$ if that will make the problem more interesting.