It is clear that the apostles invoked the name of Jesus when baptizing converts. Acts 2:38; 8:16; 10:48; 19:5; 22:16 ("calling" or "invoking"). See also 1 Cor 1:13; 6:11; Rom 6;3-4; Gal 3:27. The Greek wording leaves no doubt. For full discussion, see my small book "In the Name of Jesus."
I thought I should comment on some aspects of these texts that David Bernard doesn't address, unless he does so in his book which I've not read. The fact is that in ALL these cases, the reference to the name of Jesus concerned either Jews, quasi-Jews or God-fearers. In Acts 2 it concerned diaspora Jews. In Acts 8 it concerned Samaritans, who while only partly Jewish, certainly accepted the Jewish view of God. In Acts 10, it was a God-fearer, a Gentile who already had attached himself and his family to the synagogue and presumably the Jewish understanding of God. In Acts 19, it was a group of people who already had accepted John's baptism, and therefore, it is to be presumed that they also had accepted the Jewish understanding of God. In Acts 22, it was Paul himself, who was certainly a Jew.
This point is extremely pertinent with regard to the 1st century when the earliest Jewish Christians were only preaching the message about Jesus to other Jews (Ac. 11:19). Jews and all those who accepted their religious framework already believed in God, the Father, the Creator of the universe who spoke to Abraham, gave the Torah to Moses and inspired the prophets. They already believed in the Holy Spirit, that mysterious presence of God who is everywhere present in the Hebrew Scriptures. What they necessarily needed to accept was the messiahship of Jesus. This was the crucial point of faith for them—that Jesus was the Messiah sent from God! Hence, for them to be baptized in connection with the name of Jesus entirely makes sense, since this was the pivotal point of their new faith that made them distinctively Christian. The use of the name Jesus was not a magic formula. Rather, it was an acknowledgement that they now had a new center of faith, and that center was Jesus the Messiah. For Jews and all those who had accepted the Jewish view of God, this was a HUGE issue, an issue big enough to distinguish them from all other Jews, proselytes and God-fearers. Interestingly, when conversions of non-Jews are described in the Book of Acts, in NONE of them is used this language of "in the name of Jesus" with respect to baptism. It simply is not there--not for Lydia (Ac. 16:15), not for the Philippian jailor and his family (Ac. 16:33) and not for the Corinthians (Ac. 18:8).
Matthew's Gospel, on the other hand, focuses on the movement of the message of Jesus from the Jews alone to the wider scope of the non-Jews. Early on, Jesus clearly voiced a restriction regarding ministry beyond the circle of Jewry during his earthly life (Mt. 10:5-6; 15:22-24). However, by the end of Matthew’s gospel, this restriction was lifted, and the preaching of the gospel was extended in the Great Commission to the nations (Mt. 28:19).
For non-Jews, the issue of faith was much broader than simply acknowledging Jesus as the Messiah. It also included the whole Jewish view of God, the one and only God who was Creator and sustainer of the universe, who by his Spirit had inspired the Hebrew Scriptures. Hence, it makes sense that Matthew offers the longer baptismal wording of "in the name of the Father and the Son and the Holy Spirit" (Mt. 28:19). This language was only appropriate for those who prior to their coming to Christian faith had worshiped the pantheon of gods and goddesses in the Greco-Roman world. This longer wording reflects a larger change in viewpoint. These non-Jewish converts needed to embrace the view of the Hebrew Scriptures and the Christian message, as St. Paul puts it:
For even if there are so-called gods, whether in heaven or on earth (as indeed there are many ‘gods’ and many ‘lords”), yet for us there is but one God, the Father, from whom all things came and for whom we live; and there is but one Lord, Jesus Christ, through whom all things came and through whom we live.
It is not surprising, then, to find that in post-apostolic literature one finds in the Didache (around the end of the 1st century and roughly contemporaneous with the writings of John) a clear instruction for baptism: Baptize in the name of the Father, and of the Son, and of the Holy Spirit (Didache 7.1.3). The Didache reflects this broader scope of the gospel to the non-Jewish nations of the world.
Another point or two may be appropriate, since Bernard wishes to cite the Greek text. The language of “invoking” (the 1st aorist middle voice participle of epikaleo) does not appear in Acts 2:38; 8:16; 10:48 or 19:5 as might seem to be implied in Bernard’s statement. These passages about Jewish baptism simply say that baptism was performed "on (epi) the name of Jesus Christ" or "into (eis) the name of the Lord Jesus" or "in (en) the name of Jesus Christ". Three different prepositions are used, which suggests that the real point was not precision of wording, but a general acknowledgement that Jesus was Messiah and Lord. The ONLY place where the language of invocation is used in the Book of Acts with respect to baptism is in 22:16, and as the middle voice participle in the Greek text makes crystal clear, the one who uttered this invocation was not the one baptizing Paul, but Paul himself as he was being baptized! It was Paul who was calling on the name of the Lord, not Ananias! Would David Bernard or those of his persuasion be comfortable if only the candidate for baptism said anything at the time of baptism, not the preacher performing the baptism? Yet this is exactly the case for Paul, and the Greek text will admit no other possibility.
Hence, Bernard's statement that the apostles "invoked the name of Jesus when baptizing converts" and that "the Greek wording leaves no doubt" is either a misunderstanding of the Greek text or disingenuous. The funny thing about "facts" is that they keep getting in the way, and in this case, they are definitely in the way of Bernard's statement that "the Greek wording leaves no doubt". The Greek wording leaves a great deal of doubt!