How make the character controller the child of another GameObject

I want to attach the first-person character controller to another object so that moving that object moves the controller.

This is the code I'm working with (I use this script on the controller itself):

 public class AttachToHead : MonoBehaviour {
      public GameObject node;
      void Start () {
     
      }
      
 void Update (){
         
             node = GameObject.Find("Cube");
             this.transform.parent = node.transform;
           
      }
      
    }

The code works for other GameObjects that I've tested, but it does not work with the controller. There most be something special about it, but I haven't been able to figure out what that is.

I have asked this question before but I didn't receive a good answer, so I'm hoping someone else can help me out.