Warning: The following feature contains SPOILERS for Black Lightning season 4, episode 6, "The Book of Ruin, Chapter Two: Theseus' Ship"
The fate of the League of Assassins in the Black Lightning. It was suggested the League would turn over a new leaf during the build-up to Crisis on Infinite Earths but now it seems the organization has returned to its roots while evolving to meet the challenges of Earth-Prime.
One of the major subplots of Black Lighting season 4 has centered around the war for control of Freeland's underworld between two gangs, The 100 and the Kobra Cartel. Latavious "Lala" Johnson, leader of The 100, proved too difficult for the Kobra Cartel to bring down, thanks to the magical curse that resurrects him whenever he dies. This prompted the Kobra Cartel to bring in special help; a hired killer called Ishmael, who was said to be an expert in killing metahumans.
The League of Assassins was mentioned in the closing moments of the Black Lightning episode "The Book of Ruin, Chapter Two: Theseus' Ship," as Ishmael presented the Kobra Cartel's acting leader, Destiny, with a concrete-encased Lala, in clear imitation of the carbonite-frozen Han Solo. Destiny inquired about keeping Ishmael on retainer, but he politely declined saying that he had other business to attend to elsewhere. Destiny revealed that she had heard about this "other business"and that she'd heard Ishmael was auditioning for a place among the League of Assassins, having been tasked with killing 100 metahumans to prove his worthiness. When Ishmael said that, counting Lala, he was only six short of the required 100 metahumans, and Destiny offered to pay him to take care of the rest of Freeland's metahuman vigilante population, including Black Lightning. While this sort of test would not be out of line for the League of Assassins in the Arrowverse, it does stand at odds with what was predicted for the organization during the final season of Arrow, heading into Crisis on Infinite Earths.
The Arrow episode "Leap of Faith" detailed Thea Queen's efforts to stop two splinter factions of the League of Assassins, which had been disbanded by Nyssa Al Ghul in Arrow Season 4, from recovering the Sword of Al-Fatih, which had once served as a symbol of rulership among the first Assassins. The episode ended with one faction destroyed and Thea besting the leader of the other faction, Talia Al Ghul, in single combat. Talia offered her allegiance and that of her followers to Thea, but she shrugged off any suggestion of restarting the League of Assassins, saying that what the world really needed was a League of Heroes. Talia agreed with the sentiment (if only because it would have angered her father) and accepted Thea's offer to build such a group together.
It's possible that the fate of the League changed completely following the events of Crisis on Infinite Earths, though it is hard to believe Oliver Queen would allow the universe to be reborn with a group he and his sister had worked so hard to destroy intact. It also seems unlikely that Talia would go back on her bargain or that Thea would forget about her plans to build a better League after she agreed to marry Roy Harper in Arrow's finale. Given all that, it seems most likely that the League of Assassins Ishmael seeks to is a new organization that has stolen the name of the original League.
It is worth noting that Ishmael is the name of a League of Assassins member created for the most recent Batman and the Outsiders comic, where Black Lightning was a major character. However, while the characters sport similar costumes, the comic book version of Ishmael was given metahuman powers as a result of experiments Ra's Al Ghul conducted using the water of the Lazarus Pit. The Arrowverse version of Ishmael doesn't seem to have any powers beyond being a skilled swordsman and master strategist. Presumably episodes of Black Lightning will fully explain Ishmael's origins and the League of Assassins' status in the Arrowverse.