The mystery character is Claude from the main series. Near the end of Liberation Day, she's revealed to be the Wanderer, an entity who can manipulate time and space to such an extent that she can affect alternate dimensions, creating and destroying different timelines at will (the entire [Re]turn DLC revolves around this). Her appearance in Academy suggests the ability to freely traverse timelines/dimensions as well, though it's possible she may be limited to following certain connections between timelines (such as Academy Sola being an "echo" of Sola from the main series that was created when she was displaced just before the Sharr'lac fired). While she notes in [Re]turn that she has to limit her time manipulation if she wants to preserve whatever timeline she's currently inhabiting, Claude is nonetheless the most powerful known individual in the Sunrider franchise, with Sola even musing that understanding the scope of her powers would require one to think of her as being closer to a god than a person.