# Is being a contranormally closed subgroup a transitive property?

This question is related to

Are contranormally closed subgroups normal subgroups?

where one can also find the definition of a contranormally closed subgroup.

Given a group $G$, a contranormally closed subgroup $H$ of $G$ and a contranormally closed subgroup $K$ of $H$, is $K$ a contranormally closed subgroup of $G$? I conjecture that this is in general not true (similar as for normal subgroups).

I think this property is transitive. Let $L$ be the contranormal closure of $K$ in $G$. Then the normal closure of $H$ in $\langle H,L \rangle$ contains $H$ and $L$, and so is equal to $\langle H,L \rangle$.
But then since $H$ is contranormally closed in $G$, we get $\langle H,L \rangle=H$, so $L \le H$, and hence, since $K$ is contranormally closed in $H$, $L=K$.