A note on $\mathcal{M}$-normal embedded subgroups of finite groups

In this note, we obtain a criterion for $p$-nilpotency of a finite group. We not only simplify the proofs of the main theorems of Chen et al. [On $\mathcal{M}$-normal embedded subgroups and the structure of finite groups, Math. Scand. 127(2021), no. 2, 243–251] but also generalize these results.