Finite groups in which normality is a transitive relation

Let G be a finite group. We say that G is a T₀-group, if its Frattini quotient group G/F(G)G/Phi (G) is a T-group, where by a T-group we mean a group in which every subnormal subgroup is normal. We determine the structure of a non T₀-group G all of whose proper subgroups are T₀-groups.