18.91 (2014)

A subgroup $H$ of a group $G$ is said to be propermutable in $G$ if there is a subgroup $B \leqslant G$ such that $G = N_G(H)B$ and $H$ permutes with every subgroup of $B$.
$\qquad$ a) Is there a finite group $G$ with subgroups $A \leqslant B \leqslant G$ such that $A$ is propermutable in $G$ but $A$ is not propermutable in $B$?
$\qquad$ b) Let $P$ be a non-abelian Sylow 2-subgroup of a finite group $G$ with $|P| = 2^n$. Suppose that there is an integer $k$ such that $1 < k < n$ and every subgroup of $P$ of order $2^k$ is propermutable in $G$, and also, in the case of $k = 1$, every cyclic subgroup of $P$ of order 4 is propermutable in $G$. Is it true that then $G$ is 2-nilpotent?