Its really difficult to say what the dead three may want as the ultimate goal - and although some kind of unlimited godhood may be a distant goal there are necessary steps toward it. We can only guess based on their previous behavior. Shar has an even bigger grudge to hold and she would also very probably prefer to evade being under the control of Ao. Although she is by all accounts also on the side of destroying the whole Toril/Faerun and everything else. One other way to evade being under Ao control is to somehow destroy him. But those are some really high end goals. Wild Speculations at best.
If we look at the current situation, Shar has been greatly weakened as were the Dead three so their first immediate concern in getting their previous powers back.
"The Dead Three, greatly reduced in power, decided to remain on Toril, living as quasi-divine mortals, to spread the word of their return, gather more worshipers, and influence events in their favor. Bane led the trio in these efforts." Shar rebuild her power and dominion in the Astral plane. She had already previously impersonated other gods, killed other gods, used Bane and Cyric to kill Mystra, caused the spell plague and so on.
It seems more or less obvious that she and the dead three are involved in the cult of the Absolute. The only question remains whether there is any other additional deity or two in the game, or if the Ilsensine is just another of Shar impersonations created to gain influence over Illithids and their special capabilities such as tadpoles present.
