Is it hard to reduce God to a checklist
"But will God indeed dwell on the earth? Behold, heaven and the highest heaven cannot contain You, how much less this house which I have built!" 1 Kings 8:27
What God cannot do
Remember the Angel's words
"For nothing will be impossible with God.” Luke 1:37
So asking what is this that God cannot do may be a hard question.
In the OT the Holy Ghost is mentioned several times with characteristics and activities that may overlap with those of the Father, but it is mentioned separately in the Scriptures as an entity (similar as the Father is mentioned).
"Create in me a clean heart, O God, And renew a steadfast spirit within me. Do not cast me away from Your presence And do not take Your Holy Spirit from me." Psalm 51:10,11
In the NT Jesus explains the role of the Holy Ghost as comforter, guide, teacher.
Jesus portrays the Father deciding and accompanying in the passing of every bird
"Are not two sparrows sold for a cent? And yet not one of them will fall to the ground apart from your Father." Matthew 10:29
A humble task for the Father.
In John 14 Jesus says
"But the Helper, the Holy Spirit, whom the Father will send in My name, He will teach you all things, and bring to your remembrance all that I said to you." John 14:26
So, in words of the Son, the Spirit is sent by the Father.
In John 15 Jesus addresses again the issue
"When the Helper comes, whom I will send to you from the Father, that is the Spirit of truth who proceeds from the Father, He will testify about Me" John 15:26
Since the idea of the Father sends the Spirit is repeated for the 2nd time in the same conversation, it may be an important one. I understand that the Father and the Spirit are two persons.
The Apostles say the Spirit is an intercessor, comforter and guide.
Paul in Romans 8 says
"In the same way the Spirit also helps our weakness; for we do not know how to pray as we should, but the Spirit Himself intercedes for us with groanings too deep for words" Rom 8:26
It is given that intercedes for us upon the Father, so there are different roles again.
Then he tells us that Jesus is interceding
"who is the one who condemns? Christ Jesus is He who died, yes, rather who was raised, who is at the right hand of God, who also intercedes for us." Rom 8:34
Paul says that Jesus is the intercessor, so there are some overlapping roles not only between the Father and the Spirit, but also between the Spirit and the Son.
The mystery of the Trinity is hard to explain for us mortals, let the Spirit guide us to the truth.