What activities would a Product Owner typically undertake in the phase between the end of...

There are no activities that a Product Owner would typically undertake in the phase between the end of the current Sprint and the start of the next Sprint, because there is no such phase. The next Sprint starts immediately after the current Sprint, without any gaps or breaks. The other options are either activities that occur during a Sprint (such as refining the Product Backlog) or activities that are not consistent with Scrum (such as working with Quality Assurance departments or updating project plans).