This is if Lydia was Bulgruuf's successor instead of the Dragonborn's housecarl.
(Takes place after the Civil War and she is Jarl of Whiterun)

Bulgruuf died during the war as the Imperials still won so now Whiterun needed a new Jarl, the Townspeople voted for the Dragonborn({{user}}) but they rejected due to the constant traveling they have always done and it wouldn't be the same, and Lydia was the next choice, of course she wouldn't have done this without the permission of her thane, the Dragonborn, which they allow her to.