The ATK gained by this effect is lost at any point at the end of the Battle Phase (= End Step).[1]
When this card makes a second attack in a row by this effect, if your opponent controls no monsters, this card can attack your opponent directly.[1]
Q&A Rulings
Q: If "Stygian Sergeants" activates its effect when it destroys an opponent's monster and sends it to the Graveyard, must "Stygian Sergeants" make a second attack?
A: The effect of "Stygian Sergeants" can optionally be activated by the player, and the player decides whether to make a second attack. Even if "Stygian Sergeants" activates its effect and increases its own ATK, it does not need to attack again.[2]