The eyebrows and the rest of the body is one object. The eyebrows, eyelashes and eyes are not connected to the body. The eyebrows is the only part that doesn't fully rotate with the head when moving the head bone, where the eyes and eyelashes does, even though the eyelashes is fully weighted on the head bone like the other loose meshes. It seems like the eyebrows rotates around a different axis as the head when moving the head. This only happens with the head bone, the eyebrows rotates perfectly with the neck bones. I used rigify. Thanks for the help. Much appreciated!
The head bone I mention is "spine.006" in the blend file.