The establishing, negotiating, and naming of power and acts of power is inherently political and very often religious.
The Jewish leaders (including elders, scribes, the High Priest and those associated with his office) had held the apostles in custody overnight, and the following day they summoned Peter and John to interrogate them. Like the leaders and the people, Peter and John are devout Jews, as well as disciples commissioned by Jesus who himself was a loyal worshipper of YHWH. The leaders want to know the source and authority of their power to heal a man crippled from birth: "By what power (dumanis) or by whose name (onoma) did you yourselves perform this act," 4:7.
Implicit in the question, it appears, is the understanding that the apostles did not perform the miracle by their own strength and a concern for the potential emergence of a rival "power," "name," or a new Jewish sect that might threaten the status quo (cf. 28:22). Also, the leaders do not assume that the miracle occurred by the power of or in the name of YHWH, despite the fact that all of them are worshipers of YHWH. Religious folks who have confused the power of position with the power of God are more likely to reject the power of God operating in others who lack similar position and rank (cf. 4:13), despite how God might use them. We should maintain some humility considering our fallibility, mortality or human condition no matter how high we might climb in institutions. Only God is infallible, inscrutable, and absolutely God.
Those same religious folks might misguidedly attribute God's miracles to some other power or name that is antithetical to God. Interestingly, the people (laos) did not ask this question but were somewhat celebrity struck by what Peter and John did for the lame man, 3:12.
This scene is reminiscent of Acts chapter 8 where Luke writes of Simon that "He was the power (dumamis) of God called great," 8:10. Both this text and our pericope may betray a bias on the part of some against what cannot be explained by our conventional understandings of how God functions in our world or against phenomena that falls outside the boundaries of our theologies. Many would not believe that God was active in and through Jesus of Nazareth; it did not fit with their constructed theologies (cf. 4:2-3).
This is the first time in Acts that Peter is explicitly described as "filled with the Holy Spirit," 4:8. Perhaps, this description serves to set Peter (and the apostles) over against the leaders and/or to authenticate the content of his speech: "Let it be known to you all that by the name of Jesus Christ of Nazareth whom you yourselves crucified and whom God raised from among the dead, by this [name] this [lame] man stands before you whole. This [Jesus] is the stone, the head cornerstone, who was reject by your builders," 4:10-11. The healing of the lame man in the name of Jesus becomes evidence for the power of God operating through his name. The name has power beyond the grave because God lifted Jesus out of grave and made him both Christ and Lord (2:36).
The phrase "full of the Holy Spirit" is a term that is never applied to women in Acts, in spite of the fact that God's Spirit fell on the women also who were at Pentecost (1:14; 2:1-4). In Acts, Luke describes women such as Dorcas, as full of good deeds (9:36; cf. Luke 1:35, 41). When the seven Hellenists (including Stephen and Philipp) are chosen for the daily table ministry as opposed to the apostles' ministry of the word of God, the seven men are selected because they are full of wisdom and of the Holy Spirit (6:3,5).
Here again, the phrase is used to distinguish one group over against others. There is danger in a single story because some stories remain untold and the telling of a story is never synonymous with the real event(s) themselves. The many stories in the Bible testify about God and Jesus (cf. John 5:39; 21:25). God cannot be circumscribed or contained within any story humans narrate, if only because human language is sufficiently insufficient.
That "God raised Jesus" was a critical kergma (proclamation) among the early believers in Jesus as God's Messiah (1 Corinthians 15). This phrase is repeated in missional speeches in Acts (4:10; 10:40; 13:34, 37). A careful read of Acts, in fact, shows that it is not the Holy Spirit (or Peter or Paul) who are the primary actors, but it is God. We have lost sight of the theocentric Jesus and the theocentricity of the Scriptures. God raised Jesus. God promised God's Spirit. God poured out God's Spirit.
A theocentric approach to religion is not a watering down of Christianity, but it is re-membering of an ancient foundational kerygma. A theocentric faith focuses on the power of God at work in and through Jesus, at work in believers, and at work in the world performing miracles or powerful acts. A theocentric faith acknowledges that God acts in ways I don't understand. God invested Jesus with power! And Jesus said we would do greater things. God raised Jesus!
Peter further declares that there is no other name under heaven given among humans by which we must be saved, 4:12. Perhaps, the people and religious leaders wondered about the salvation of their relatives, the patriarchs and matriarchs, and the many, many people who had lived and died before Jesus of Nazareth ever walked the earth. What was undeniable was the evidence of God at work in the world: The man who had been lame from birth stood before them whole, 4:14. Does our faith require that we have all the answers? God, thank God, continues to work in the world even while we struggle to understand and to articulate, imperfectly, our faith in God.