Aura seems to be missing in all if not the majority of the //G.U. anime and games, //Quantum, but makes some appearances in the //Beyond the World movie?
What happened to her between this period of time?
In The world:R2, Aura is missing because she is trying to let the players act by themselves. She acted tough by creating the 3 azure knights (azure kite, balmung and orca) and helped haseo during the last fight of .hack//G.U. game series.
After Aura was reborn as the Ultimate AI during the events of .hack (particularly the final battle against Morganna Mode Gone's manifestation in Corbenik's final form) she was referred as a Goddess in a world which was heavily reliant on the world wide network to which she was the ultimate debugger.
Now i'm not sure which .hack//Legend of the Twilight is cannon however before it she gave birth to her daughter Zefie, created a false competition to recreate BlackRose and Kite PC's and gave them to Rena and Shugo and gave Shugo the Bracelet (probably only because he had the Kite PC), after that
In the Anime she helps Shugo and Rena by saving Rena after she is killed and helps them in stopping Morti's Skieth (not entirely sure if it's a recreated Skieth but it looks a hell lot like it)
In the Manga Zefie is escorted back to Aura by Shugo and Rena
sometime after Legend of the Twilight she began to dislike this role as "goddess" (Endrance comments on this in Net Slum in the G.U Games) and suddenly disappeared by going into a deep sleep. The sudden disappearance had an affect on the world wide network and CC Corp saw this as the perfect opportunity to create a new Aura which they would control (thus restoring their and ALTIMIT's market hold)
Project G.U which was supposed to duplicate the scenario that lead to Aura's creation, however failed when the R.A (Rebirth Aura) Program failed to run and ended up wiping out most of The World's data. CC Corp them merged the remaining data of The World with another game they was working on to create The World:R2. before The World's Shutdown player were given some time to explore what was left with their PCs. Reiko Saeki's (Pi from G.U, Shamrock from Quantum) older bother Jun Bansyoya who was apart of Project G.U encountered Zefie where she tells him how Aura abandoned her position as the Ultimate AI and integrated into the system of The World, afterwords she herself "returned to the winds." (possibly to return to her mother)
After the AIDA began to appear and become influenced by Tri-Edge, Aura wakes up creates the Azure Knights to protect The World however they was rushed in their creation so their AI was incomplete so to "protect The World" meant to "Protect Aura", anyone who approached Key of The Twilight would also become their target as the key was an extension of Aura. When Hasao fought the Knights, Aura orders the Knights to stop fighting using Aina's PC as a medium, telling them how she's leaving the future of The World in their hands (the players) and they had to stop Cubia themselves. however she does given them some assistance by creating Aura Spheres within Cubia and after Cubia is destroyed she gives Hasao the Member Addresses of the Knights and a Card before returning to sleep, so she's not entirely without power.
I'm sure that .hack//LINK occurs before .hack//Beyond the World but i'm not entirely sure on the story of it as it's a manga and game, in the game Aura has woken again and near the end becomes infected with a Virus which makes her become the crazed Queen of Demise and in a twisted form of love tried to Real digitalize humanity, she's saved by AIKA's sacrifice and AIKA presumably returned to the sea of data where Aura sleeps with Aura as she herself falls back to sleep
that's all i know without properly looking into .hack//Link but for the most part she in a deep sleep in the sea of data and only wakes up with The World is in danger and while caring for the players in it, tries to aid them without directly interfering so that the fate of The World and the players are decided by the players and not by her