On the product of a $B$-group and a primary group

A finite non-nilpotent group $G$ is called a $B$-group if every proper subgroup of the quotient group $G/\Phi(G)$ is nilpotent. The main properties of $B$-groups are established and the group factorized by a primary and a $B$-group is studied. In particular, it is proved that if $G=HK$ is the product of a $B$-subgroup $H$ with a primary subgroup $K$, and if the order of the non-normal Sylow subgroup of $H$ is not equal to $3$ or $7$, then $G$ is solvable.