Influence of properties of maximal subgroups on the structure of a finite group

We establish some tests for the solvability of finite groups and describe one class of unsolvable groups. We prove that an unsolvable group $G$ such that a maximal subgroup $M=P\times H$ is nilpotent and the 2-Sylow subgroup $P$ of $M$ is metacyclic has a normal series $G\supseteq G_0\supset T\supseteq\{1\}$ such that $T$ is contained in $M$, $G_0/T\simeq PSL(2,q)$, where $q$ is a power of a prime of the form $2^n\pm1$ and the index of $G_0$ in $G$ is not greater than 2.