Class of nilpotency of semidirect product

Question

Let $G=N\rtimes H$ for some groups $N$ and $H$. Is it true that the nilpotency class of $G$ is at most maximum of nilpotency classes of $N$ and $H$, if it is known that both groups are nilpotent? And if not is it true, that the nilpotency class of $\mathrm{Hol}(K)$ is at least $n$, where $n$ is nilpotency class of $K$, where $K$ and $\mathrm{Aut}(K)$ are both nilpotent? I tried to check the last statement in SAGE and it holds for all the groups I checked by now.