Conjugate-permutability is conjugate-join-closed

Statement with symbols
If $$H$$ is a conjugate-permutable subgroup of a group $$G$$, and $$S$$ is any subset of $$G$$, then the subgroup:

$$H^S := \langle H^g \mid g \in S \rangle$$

is also a conjugate-permutable subgroup of $$G$$.

Similar facts

 * 2-subnormality is conjugate-join-closed
 * Normality is strongly join-closed
 * Permutability is strongly join-closed

Applications

 * Maximal conjugate-permutable implies normal