Auto Activate should be disabled while refreshing to avoid auto activation of sandbox. The old sandbox will be available until we click Activate button after the refresh.

Activating a refreshed sandbox replaces the current sandbox with the refreshed version and permanently deletes the older version and all data in it. Your production org and its data aren’t affected.

Click Discard to discard a refreshed sandbox. This action is displayed only for sandboxes that aren’t activated. You can discard a sandbox if you don’t want to use it or if you plan to do another refresh.