Strictly speaking, an automated activity does not have a human actor as role involved, and from that perspective the automated one does not belong in the swim-lane of that role, but in the one of the "automatic" role instead.
You may say that this is smells like over-designing the thing, and moreover it takes extra space, so rather not.
But there also is a practical edge to this. I happened to notice that there are certain circumstances in which automatic activity can actually show up in the workspace of people having that role. That can happen when the automatic activity takes some time to complete, and (if memory serves me well) I also saw this sometimes happening with timers. Very confusing for the user, because it looks like he or she has something to do, while the workspace does not allow doing so.
I therefore recommend always to put automatic activities in the "automatic" role, at least once you start implementing the process. The obvious exceptions being automatic activities inside a screen-flow.