Should the search-box that was used to search for a user hold it’s search results upon return to address book after editing a user?

When editing a user's info and clicking Save, the page navigates back to address book. Should the search-box that was used to search for that user in the first place "hold" its search results - displaying only the search results upon returning to the main user list - or should it show the full list of users again?

I am wondering about this because I am using table data throughout a software application that has this functionality happening to more than just editing user's info. (searching and editing a Customer's Account, Locations, etc.)