On the permutability of Sylow subgroups with derived subgroups of $B$-subgroups

A finite non-nilpotent group $G$ is called a $B$-group if every proper subgroup of the quotient group $G/\Phi(G)$ is nilpotent. We establish the $r$-solvability of the group in which some Sylow $r$-subgroup permutes with the derived subgroups of $2$-nilpotent (or $2$-closed) $B$-subgroups of even order and the solvability of the group in which the derived subgroups of $2$-closed and $2$-nilpotent $B$-subgroups of even order are permutable.