Batman's actions against Red Hood, despite him secretly being on Batman's side, could jeopardize any chance of them working together again after "Gotham War."

Source: https://www.cbr.com/red-hood-catwoman-batman-gotham-war-true-mission/

He's been shown, at times, as being the original Red Hood before an acid bath warped his mind.

Source: https://www.cbr.com/batman-lore-important-dc/

However, whether Batman is being controlled or not, Zur-En-Arrh is still a part of him, and it's unlikely Red Hood will be willing to forgive Bruce anytime soon.

Source: https://screenrant.com/red-hood-change-batman-no-kill-rule/

It's up to Batman's Outsiders and Red Hood to figure out how to infiltrate the event and get the secrets before anyone else can.

Source: https://www.cbr.com/dc-batman-legends-of-gotham-1-review/

Red Hood and Robin had the benefit of sharing a nightmare, but their fortune was not to last.

Source: https://screenrant.com/knight-terrors-dc-creepiest-event/

Since the story has been unclear about the true identity of the Joker, it's entirely possible Red Hood is teaming up with Joker to kill someone else.

Source: https://screenrant.com/joker-team-up-red-hood-kill-imposter/

Still seeking vengeance for Bane's murder of Alfred Pennyworth, Red Hood gladly takes the opportunity.

Source: https://www.cbr.com/red-hood-outsiders-team-up-batman-dc-comics/

Tim Drake suggests studying Selina's method and Red Hood even volunteers to be Catwoman's inside man.

Source: https://screenrant.com/batman-recruiting-riddler-bat-family/

Unsurprisingly, the web-comic claimed the titular Red Hood was the most like Batman, citing his rage and aggression towards criminals as being the primary features that he shared with Batman.

Source: https://www.cbr.com/batman-red-hood-similarities-dc/