Oliver gave up everything and is obviously struggling with fulfilling his deal with The Monitor; both The Monitor and a close friend are keeping a huge variable to themselves, and Lyla is more or less actively and effectively lying.
In this sense, Lyla did betray them, although her role as Harbinger in the “Crisis on Infinite Earths” crossover always guaranteed a connection with The Monitor.
Lyla did something that was the opposite of what Oliver intended, and she’s keeping an important relationship with The Monitor a secret from Oliver and her husband.