Class preserving automorphisms of groups

Let $G$ be a group and let $\operatorname{Aut}_{c}(G)$ be the group of the class preserving automorphisms of $G$. We prove the following: (i) If $G$ is (nilpotent of class $c$)-by-(soluble of derived length $d$), then $\operatorname{Aut}_{c}(G)$ is (nilpotent of class $\leq c-1$)-by-(soluble of derived length $d+1$ or $d$), which extends a result of Rai. (ii) If $G$ is a $B_{1}$-group, then $\operatorname{Aut}_{c}(G)$ is (nilpotent of class $\leq n-1$)-by-soluble, where $n$ is the length of a finite chain of $G$.