Hate and empathy are unrelated. To paraphrase it’s possible to hate the game not the player.
The WWI Christmas truce in 1914 shows this dynamic very clearly. It’s likely these people had great empathy for each other as they where in similar situations even sleeping in the same trenches as the front moved around. Yet, eventually they went back to killing each other. https://en.m.wikipedia.org/wiki/Christmas_truce
Those war victims didn't necessarily hate each other. They killed because they had to. Those who were subtle in their reluctance to kill were more likely to be killed in battle. Those who weren't subtle were court-martialed and executed.
I wouldn't consider hate and empathy mutually exclusive personally (e.g. you can empathise with somebody that was bullied by their parents, but still hate them because they bully their children), but I do think that somebody that derives pleasure from the suffering of someone that they hate is unempathetic.
"Hate the sin, not the sinner."