On finite groups in which every non-abelian subgroup is a TI-subgroup or has $p'$-order

We obtain a complete characterization of the structure of a finite group $G$ in which every non-abelian subgroup is a TI-subgroup or has $p'$-order for some fixed prime divisor $p$ of $|G|$ (that is every non-abelian subgroup of $G$ of order divided by $p$ is a TI-subgroup for some fixed prime divisor $p$ of $|G|$).