On $p$-locally N-closed formations of finite groups

All groups considered are finite. A formation $\mathfrak{F}\ne\emptyset$ is called locally N-closed (N-closed) in a group class $\mathfrak{X}$, if the following assertion holds: if $G\in\mathfrak{X}$ and $P\le Z_{\mathfrak{F}}(N_G(P))$ ($N_G(P)\in \mathfrak{F}$ respectively) for every Sylow subgroup $P\ne1$ in $G$, then $G\in\mathfrak{F}$. It is proved that in the soluble universe, every hereditary saturated locally N-closed non-empty formation is N-closed. It is proved that the formation of all supersoluble groups is N-closed in the class of all soluble groups with $p$-length $\le1$ for every prime $p,$ and is not N-closed in the class of all soluble groups with $p$-length $\le2$ for every prime $p$. The authors also consider $p$-locally N-closed formations.