Recall that a subgroup $H$ of a group $G$ is called a TI-subgroup if for every $g \in G$ it is $H \cap H^g \in \{1,H\}$. We say that a group $G$ is a DTI-group if the derived subgroups of all of its subgroups are TI-subgroups.

Question: Is it true that a non-trivial finite perfect DTI-group is necessarily simple?

Remark: I know that -- provided that it exists -- a non-trivial finite non-simple perfect DTI-group necessarily is minimal non-solvable.

Related question: References on a certain generalization of Dedekind groups

Recall that a subgroup $H$ of a group $G$ is called a TI-subgroup if for every $g \in G$ it is $H \cap H^g \in \{1,H\}$. We say that a group $G$ is a DTI-group if the derived subgroups of all of its subgroups are TI-subgroups.

Question: Is it true that a non-trivial finite perfect DTI-group is necessarily simple?

Remark: I know that -- provided that it exists -- a non-trivial finite non-simple perfect DTI-group necessarily is minimal non-solvable.

Related question: References on a certain generalization of Dedekind groups