The Great takes some major liberties with the historical character of Elizaveta. In the show, she's the promiscuous, ageing "party aunt," but in reality, she was an empress in her own right and, more importantly, was not alive during Peter's reign because he succeeded her to the throne. Elizaveta was a daughter of Peter the Great who took the throne in a largely bloodless coup in 1741, overthrowing her infant cousin Ivan VI and his mother, the regent Anna.
Montefiore describes how Elizaveta's reign was remembered kindly for her refusal to sentence anyone to death and her popular domestic reforms, although she also was something of a luxurious spendthrift. She left the throne to her nephew Peter III, who was her sister Anna's son (a different Anna than the regent she overthrew), and died in 1762.