Is it permissible to kill a person who assaulted one's father with a stick, causing a serious head injury, out of fear that he may repeat his assault, which could lead to death? And is it permissible to pursue him to kill him if he flees after the assault?

The man's action is a great evil (munkar), involving oppression and aggression. It is not permissible for a Muslim to harm his brother. It is also not permissible to kill him, even if he killed your relative, because the implementation of [Sharia] punishments (hudud) is for the Muslim ruler or his deputy, not for the general public, out of fear of chaos.

His case must be reported to the responsible authorities to deter him, reprimand him, and apply the necessary punishment. If he aggresses again, he is an aggressor (sail) who should be repelled by the easiest means first, then the next easiest. It is not permissible to kill him if he can be repelled without killing him.