Let $G$ be a group. A subgroup $H$ of $G$ is called core-free if \begin{equation*} \bigcap_{g\in G}gHg^{-1} = 1. \end{equation*} Equivalently, $H$ is core-free if and only if for any subgroup $N$ of $H$, if $N\lhd G$ then $N = 1$.
Now assume groups below are finite.
A Dedekind group $G$ (a group such that all its subgroups are normal) has no nontrivial core-free subgroup as \begin{equation*} \bigcap_{g\in G}gHg^{-1} = H\ne 1 \end{equation*} for any nontrivial subgroup $H$ of the Dedekind group $G$. This list includes all abelian groups and Hamilton groups ($Q_8\times A\times B$ with $A$ abelian of odd order and $B = \mathbb{Z}_2^n$ for some $n$).
My question:
I aim to characterize all finite groups which has no nontrivial core-free subgroups. An example of such group which is not Dedekind is the dicyclic group of order $12$ (see comments here). Is there any sort of classification of such groups?
Added: The motivation of this question is to find some conditions of subgroups such that quasi-primitivity implies primitivity:
Proposition: If $G$ has a regular subgroup $H$, then $G$ is quasi-primitive if and only if $G$ is primitive.
Note that if $H$ is transitive and has no non-trivial core-free subgroup, then $H$ is regular. The question comes to characterize what $H$ is up to isomorphism.
Added (Sep 10, 2019):
This is true:
$G$ has no nontrivial core-free subgroup then if all of the Sylow subgroups of $G$ are not core-free, and inparticular all of the cyclic subgroups of prime order of $G$ are normal.
For example, $C_3:C_4$ (dicyclic, $3$-Sylow is normal and $2$-Sylows contain the center) and $C_7:C_9$ ($7$-Sylow is normal and $3$-Sylows contains the center) both have no nontrivial core-free subgroups.
Also I think the following might be true:
$C_{p^d}:C_{q^e}$ has no nontrivial core-free subgroups if $(p-1)\nmid q^e$, where $p,q$ are distinct primes. In this case $p$-Sylow is normal and $q$-Sylow contains the center.