Hi- I know it would seem logical to post that question in this forum, but Rose of Sharon is actually not a member of the rose family- it's a kind of hibiscus. If I recall correctly, that family roots fairly easily from cuttings. But you might want to try one of the other forums to find someone with more direct experience.

I have rooted it from cuttings. I treat the hibiscus cuttings just like rose cuttings ... rooting hormone and planting them in a milk jug covered with the top of a soda bottle. I have good success this way.