On $\mathscr F_h$-supplemented subgroups of finite groups

Suppose that $\mathscr F$ is a formation of finite groups. We introduce the concept of $\mathscr F_h$-supplemented subgroups and investigate the structure of finite groups on assuming that some maximal subgroups of Sylow subgroups, maximal subgroups, minimal subgroups, and $2$-maximal subgroup are $\mathscr F_h$-supplemented, respectively. Some available results are generalized.