The existing rig doesn't move/deform the head.
Video: https://drive.google.com/file/d/1RM5W4aen0HiJqpmfWP6yiaHMy5S-9uL1/view?usp=sharing
Blender project: https://drive.google.com/file/d/1TcJiX2N7iK_e6G54ZoxmBmtakF2zZUQo/view?usp=sharing
I used Mixamo, so it changed the bones.
Here is original project: https://drive.google.com/drive/folders/1MnWMpi-9CXsfRhJ0dvfKhWYSIORnLx--?usp=sharing