On two questions from the Kourovka Notebook concerning maximal subgroups

Let $p$ be a prime number. When $p$ is odd, we study finite groups in which every maximal subgroup is either non-abelian simple or $p$-nilpotent, as well as those in which every maximal subgroup is either non-abelian simple or $p$-decomposable. We prove that every non-simple, non-solvable group satisfying these criteria is $p$-nilpotent, and $p$-decomposable, respectively. This answers two open questions posed by V.S. Monakhov and I.N. Tyutyanov in the Kourovka Notebook. Additionally, if $p=2$, we improve the main result of Monakhov and Tyutyanov by providing a complete classification of non-solvable groups whose maximal subgroups are either non-abelian simple or $2$-nilpotent.