Doing a quick google search - The employer can be exempt from paying their share of FICA (if they file the appropriate IRS form) which just means the employee needs to pay both parts as they are self employed. The rules are a little more complex for employee ministers. And those rules are separate from the rules for other employees of the church. There are a few ways non-minister employees can get out of it, but usually if they aren't W2'ed (and reading I was surprised to see that it was unusual that the one church dad was a minister at for 20 years until he retired he was W2'ed as that isn't common- don't know about the earlier churches he worked at and how they did it) they are then considered self employed and are bound by self employment rules unless they meet those uncommon circumstances and then choose not to pay into the "system".
Of course what is relevant to the situation this thread is about is that she didn't pay FICA for whatever reason so has an "issue" with respect to medicare.
