Supersolvable subgroups of order divisible by 3

We determine the structure of the finite non-solvable groups of order divisible by $3$ all whose maximal subgroups of order divisible by $3$ are supersolvable. Precisely, we demonstrate that if $G$ is a finite non-solvable group satisfying the above condition on maximal subgroups, then either $G$ is a $3'$-group or $G/{\bf O}_{3'}(G)$ is isomorphic to ${\rm PSL}_2(2^p)$ for an odd prime $p$, where ${\bf O}_{3'}(G)$ denotes the largest normal $3'$-subgroup of $G$. Furthermore, in the latter case, ${\bf O}_{3'}(G)$ is nilpotent and ${\bf O}_2(G)\leq {\bf Z}(G)$.