I am trying to construct an example of a group $G$ and a subgroup $H$ where the 1-hypernormalizer and the 2-hypernormalizer of $H$ are distinct.
The normalizer of $J \leq G$ is $N(J)=\{g \in G \mid gJg^{-1}=J\}$. The k-hypernormalizer of $H$ is the k-th iteration of $N(-)$ applied to $H$.
I tried but was not able to find concrete examples in a literature search (one book that comes up in references but that I unfortunately don't have is 'A Course in the Theory of Groups' by Robinson).
I have thought about finding a non-normal subgroup of a group with the normalizer condition: $J \lneq N(J)$ for all subgroups $J \leq G$. If $G$ is finite, then $G$ satisfies the normalizer condition if $G$ is nilpotent (I think also only if). However, I don't have much experience with such groups, so I am looking for some directions to follow that may be useful.