Local definitions of formations of finite groups

A problem of constructing of local definitions for formations of finite groups is discussed in the article. The author analyzes relations between local definitions of various types. A new proof of existence of an $\omega$-composition satellite of an $\omega$-solubly saturated formation is obtained. It is proved that if a non-empty formation of finite groups is $\mathfrak X$-local by F\"{o}rster, then it has an $\mathfrak X$-composition satellite.