Subnormality is not permuting upper join-closed

Statement
It is possible to have a group $$G$$, a subgroup $$H$$ of $$G$$, and intermediate subgroups $$K_1$$ and $$K_2$$ of $$G$$ containing $$H$$ such that $$H$$ is a 2-subnormal subgroup in both $$K_1$$ and $$K_2$$, $$K_1K_2 = K_2K_1$$ (i.e., they are permuting subgroups), but $$H$$ is not a subnormal subgroup of $$K_1K_2$$.

Related facts

 * Subnormality is permuting upper join-closed in finite
 * Subnormality is not upper join-closed
 * 2-subnormality is not upper join-closed