On some formations closed under taking wreath products

We construct some series of subgroup-closed saturated formations $\mathfrak{F}$ satisfying the following properties: 1) $\mathfrak{F}$ is a proper subformation of $\mathfrak{E}_\pi,$ where $\pi=\mathrm{char}(\mathfrak{F});$ 2) if $G\in\mathfrak{F},$ then there exists a prime $p$ (depending on the group $G$) such that the wreath product $C_p\wr G$ belongs to $\mathfrak{F},$ where $C_p$ is the cyclic group of order $p.$ Thus an affirmative answer is obtained to Problem 18.9 from The Kourovka Notebook.