Normal subgroups with an additional property

Consider a normal subgroup $H\lhd G$, and let $g\in G$ be some element.

In some cases, there exists a constant $k\in\mathbb Z$, s.t. $hg=gh^k$ for every $h\in H$. This $k$ can be different for different elements $g$, so denote it by $k_g$.

I'm interested in normal subgroups with the property that every element $g$ has such $k_g$.

For example:

• In the center of every group $Z(G)\lhd G$, $hg=gh$ for all $h\in Z(G),g \in G$, hence $k_g=1$ for all $g\in G$.
• Consider the dihedral group $D_n=\langle\sigma,\tau|\sigma^n=\tau^2=\tau\sigma\tau\sigma=1\rangle$ with its cyclic normal subgroup $\langle\sigma\rangle\lhd D_n$. It holds $\sigma^i\tau=\tau \sigma^{-i}$, hence $k_g=-1$ for all $g\in\tau\langle\sigma\rangle$ and $k_g=1$ for $g\in\langle\sigma\rangle$.

Is there some characterization of groups where this property holds?

• This will always hold for any cyclic subgroup (which is normal), since the condition can be rewritten as $g^{-1}hg = h^k$. – Tobias Kildetoft Mar 21 '18 at 11:38
• - Thanks Tobias! – user269559 Mar 21 '18 at 19:26