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 $H$, $L=K$.
Incidentally, for finite groups I think that a subgroupis contranormally closed if and only if it is subnormal.