On the definability of the group $L_2(7)$ by its spectrum

For a group $G$, denote by $\omega(G)$ the spectrum of $G$, i.e., the set of its element orders. We prove that every group $G$ with $\omega(G)\subseteq\omega(L_2(7))=\{1,2,3,4,7\}$ in which the product of every two involutions is a $2$-element contains a normal $2$-subgroup with primary quotient. We also reduce the investigation of groups $G$ with $\omega(G)=\omega(L_2(7))$ to that of groups generated by involutions.