Showing Subcategories instead of posts in Parent

I’ve taken a look around google, and maybe I’m just missing something, but here’s my problem.

I’m trying to show a parent category’s children INSTEAD of posts on a category page. I want them to look like a list of posts but direct visitors to the sub-category where the posts actually are.

They should display a featured image and the category description, and (oh please oh please oh please) possibly show how many posts are in that sub-category. I’m hoping this will auto-update as new sub-categories are added.

My parent categories do not have any posts, and my sub-categories will not have any children of their own.

Is there any way to do this? Is there a plug-in out there?