We know that if $H\leq G$, then $H^{G}=\left\langle h^{g};h\in H\text{ and }g\in G\right\rangle \trianglelefteq G,$ is the normal closure of $H$ in $G.$
Usually, when we kill $T\trianglelefteq G$ in $G/T$ , we have some property in $G/T.$
Example: If $T$ contains all the commutators of $G,$ then $G/T$ is abelian.
My question is: what can we say about the group $G/H^{G}$? What important properties does it have?