Some Remarks on Super $M_p$-groups

Let $G$ be a finite group and $p$ be a prime divisor of $\mathrm{\mid }G\mathrm{\mid }$. An irreducible $p$-Brauer character $\phi$ of $G$ is called super-monomial if every primitive $p$-Brauer character inducing $\phi$ is linear. The group $G$ is said to be a super $M_p$-group if every irreducible $p$-Brauer character of $G$ is super-monomial. In this note, we investigate the conditions under which a finite group $G$ qualifies as a super $M_p$-group. We demonstrate that every normal subgroup of a super $M_p$-group of odd order is an $M_p$-group.