Jason Todd (Earth-2731)

"Do I look like Batman to you? Rethink your answer before I fill you up with lead."

- Red Hood, to a Black Mask henchman

Since Jason Todd changed the Arkham Knight persona, he was able to lay his long-rooted vengeance against Batman to rest, and began to heal the damage that Joker inflicted on him at Arkham Asylum. It was during this recovery that the Red Hood was born, and is implied to have become the permanent holder of the identity. However, while he was once again aligned with his former mentor in wanting to rid Gotham City of its criminal contingent, Todd's willingness to take lives will forever keep them at odds.

Source

 * https://arkhamcity.fandom.com/wiki/Red_Hood