I was solving a problem and in the middle, I came across this. I will be really grateful for any help here. In the following, we fix the integer $s\ge 2$.

Let $U\le \operatorname{GL}(2^s,\mathbb Z)$ be a finite $2$-group. Let $N=N_{\operatorname{GL}(2^s,\mathbb Z)}(U)$ and $C=C_{\operatorname{GL}(2^s,\mathbb Z)}(U)$ be the normaliser and centraliser respectively. Suppose $N$ is acting on a set $S$, in particular $S\le \mathbb Z^{2^s}$ and action of $N$ is defined via "right multiplication".

My question is:

If we know the orbits in $S$ under the action $C$, then is there any result (or any method) which describes the orbits under $N$? Does the famous $N/C$ Theorem help in any way?

Any reference or hint will be greatly appreciated.

Thanks in advance.

I was solving a problem and in the middle, I came across this. I will be really grateful for any help here. In the following, we fix the integer $s\ge 2$.

Let $U\le \operatorname{GL}(2^s,\mathbb Z)$ be a finite $2$-group. Let $N=N_{\operatorname{GL}(2^s,\mathbb Z)}(U)$ and $C=C_{\operatorname{GL}(2^s,\mathbb Z)}(U)$ be the normaliser and centraliser respectively. Suppose $N$ is acting on a set $S$, in particular $S\le \mathbb Z^{2^s}$ and action of $N$ is defined via "right multiplication".

My question is:

If we know the orbits in $S$ under the action $C$, then is there any result (or any method) which describes the orbits under $N$? Does the famous $N/C$ Theorem help in any way?

Any reference or hint will be greatly appreciated. Thanks in advance.