Why this inner group, present at so many of the most important events in the Earthly ministry of the Lord Jesus?
He went up the mountain and summoned those whom He wanted and they came to Him.
He appointed twelve (whom He also named apostles) that they might be with Him and He might send them forth to preach and to have authority to drive out demons: (He appointed the twelve:) Simon, whom He named Peter; James, son of Zebedee, and John the brother of James, whom He named Boanerges, that is, sons of thunder; ..."
When Simon Peter saw this, he fell at the knees of Jesus and said, "Depart from me, Lord, for I am a sinful man." For astonishment at the catch of fish they had made seized him and all those with him, and likewise James and John, the sons of Zebedee, who were partners of Simon. Jesus said to Simon, "Do not be afraid; from now on you will be catching men." When they brought their boats to the shore, they left everything and followed him.
Mark 5:22-24, 35-42
One of the synagogue officials, named Jairus, came forward. Seeing him he fell at his feet and pleaded earnestly with him, saying, "My daughter is at the point of death. Please, come lay your hands on her that she may get well and live." He went off with him, and a large crowd followed Him and pressed upon Him. ... While He was still speaking, people from the synagogue official's house arrived and said, "Your daughter has died; why trouble the teacher any longer?" Disregarding the message that was reported, Jesus said to the synagogue official, "Do not be afraid; just have faith." He did not allow anyone to accompany Him inside except Peter, James, and John, the brother of James. When they arrived at the house of the synagogue official, He caught sight of a commotion, people weeping and wailing loudly. So He went in and said to them, "Why this commotion and weeping? The child is not dead but asleep." And they ridiculed Him. Then He put them all out. He took along the child's father and mother and those who were with Him and entered the room where the child was. He took the child by the hand and said to her, "Talitha koum," which means, "Little girl, I say to you, arise!" The girl, a child of twelve, arose immediately and walked around. (At that) they were utterly astounded.
Matthew 17:1-8, Mark 9:2-8
After six days Jesus took Peter, James, and John his brother, and led them up a high mountain by themselves. And he was transfigured before them; his face shone like the sun and his clothes became white as light. And behold, Moses and Elijah appeared to them, conversing with him. Then Peter said to Jesus in reply, "Lord, it is good that we are here. If you wish, I will make three tents here, one for you, one for Moses, and one for Elijah." While he was still speaking, behold, a bright cloud cast a shadow over them, then from the cloud came a voice that said, "This is my beloved Son, with whom I am well pleased; listen to him." When the disciples heard this, they fell prostrate and were very much afraid. But Jesus came and touched them, saying, "Rise, and do not be afraid." And when the disciples raised their eyes, they saw no one else but Jesus alone.
Then they came to a place named Gethsemane, and he said to his disciples, "Sit here while I pray." He took with him Peter, James, and John, and began to be troubled and distressed. Then he said to them, "My soul is sorrowful even to death. Remain here and keep watch."