The immutability is the teaching that God does not change in His nature. His character, knowledge, and existence are without variation. God has always been everywhere all the time. He has always been holy, divine, omniscient, omnipresent, omnipotent, and pure. These things cannot change in God since they are aspects of His nature. But, how do we reconcile the incarnation of the Word (which was God) and the doctrine of God's unchangeableness (His immutability)? The answer lies in understanding the doctrines of the Trinity, the incarnation, and the relationship of the two natures of Christ.
The doctrine of the Trinity states that one God exists in three persons: Father, Son, and Holy Spirit. The Son (Jesus) is the Word made flesh. "In the beginning was the word and the word was with God and the word was God . . . 14and the word became flesh and dwelt among us . . . ," (John 1:1, 14). The Bible teaches us that Jesus has two natures: God and man. This is known as the doctrine of the hypostatic union. Jesus is still both God and man, divine and human, at the same time. Jesus, as one person, exists with two natures. The divine nature "joined" with the human nature in the one person of Christ. The divine nature did not change at all in this "joining." Now, please note that the divine nature did not combine with the human nature and form a new nature called the god-man nature. That is known as monophycitism and is incorrect. The two natures are "in communication" with each other, and the attributes of each nature are ascribed to the single person. This is called the communicatio idiomatum. This Latin phrase means, "communication of the properties." In other words the one person of Christ "claims" the attributes of each nature. Here is proof: John 17:5 says, "And now, glorify Thou Me together with Thyself, Father, with the glory which I had with Thee before the world was." Notice that the man Jesus is laying claim to pre-existence and glory that He had with the Father before the foundation of the world. This is because Jesus, the person, has two natures, divine and human, and the attributes of the divine nature were ascribed to the single person of Christ.
The Word did not change
The Word did not change by adding anything to its nature. It simply joined with the human nature in the person of Christ so that two distinct natures exist simultaneously in Jesus. This is why Paul says, "In Him all the fullness of deity dwells in bodily form," (Col. 2:9). Besides, in order for God's immutability to be in question, the Word would have had to change itself by melding with the human nature into a new third thing. This would mean that it would no longer be "the Word." It would be "the WordMan" nature which is neither divine nor human but a new third thing. Furthermore, the doctrine of the incarnation denies any change in the divine Word at all. It simply states that the Word became flesh (not meaning it changed its nature). The Word resides in the person of Christ along with the human nature so that Jesus has two distinct natures.
Therefore, we can conclude that the Godhead participates in humanity through the incarnation of Christ, but the Godhead is not changed in any way.