On the class of groups with pronormal Hall $\pi$-subgroups

Given a set $\pi$ of prime numbers, we define the class $\mathscr P_\pi$ of all finite groups in which Hall $\pi$-subgroups exist and are pronormal by analogy with the Hall classes $\mathscr E_\pi$, $\mathscr C_\pi$ and $\mathscr D_\pi$. We study whether $\mathscr P_\pi$ is closed under the main class-theoretic closure operations. In particular, we establish that $\mathscr P_\pi$ is a saturated formation.