Exercise 4, § 4

Let $H$ be a nilpotent group of class $c$ and let $G$ be the group of automorphisms of $H$ which act trivially on $H/(H, H)$. Show that $G$ is nilpotent of class $\leq c - 1$. (Apply Exercise 3 to the lower central series of $H$ and note that $G$ operates trivially on $\operatorname{gr}(H)$.) Show that, if $H$ is a finite $p$-group ($p$ prime), so is $G$ (same method).