On the solvability of some finite primitive groups

Let $M$ be a subgroup of a finite group $G$ and $\operatorname{Core}_{G}M$ is the largest normal subgroup of $G$ contained in $M$. We determine the structure of the finite group $G$ if $G$ possesses a maximal subgroup $M$ with $\operatorname{Core}_{G}M = 1$ and all maximal subgroups $H$ of $G$ with $\operatorname{Core}_{G}H = 1$ satisfy certain properties.