Make method async : should "pull" await in the caller when possible

Brief description:

I am not sure I am in the correct place, so feel free to redirect me.

What I'd like when I use the refactoring "make a method async", is that all callers should automatically await my method (and propagate again to their callers if it was not alredy async, recursively)

Also, I understand that is is a convention to add "Async" suffix, but it will be ideal if it can be configured somewhere (because, at the moment, I just need to rename the method after it has been made async, but with recursion it can be a lot of renames ...)

Languages applicable:

C# (I don't know if it applies to other languages)