This is a good article on the verse:
Acts 2:38
Repent ye (μετανοησατε [metanoēsate]). First aorist (ingressive) active imperative. Change your mind and your life. Turn right about and do it now. You crucified this Jesus. Now crown him in your hearts as Lord and Christ. This first. And be baptized every one of you (και βαπτισθητω ἑκαστος ὑμων [kai baptisthētō hekastos hūmōn]). Rather, “And let each one of you be baptized.” Change of number from plural to singular and of person from second to third. This change marks a break in the thought here that the English translation does not preserve. The first thing to do is make a radical and complete change of heart and life. Then let each one be baptized after this change has taken place, and the act of baptism be performed “in the name of Jesus Christ” (ἐν τῳ ὀνοματι Ἰησου Χριστου [en tōi onomati Iēsou Christou]). In accordance with the command of Jesus in Matt. 28:19 (εἰς το ὀνομα [eis to onoma]). No distinction is to be insisted on between εἰς το ὀνομα [eis to onoma] and ἐν τῳ ὀνοματι [en tōi onomati] with βαπτιζω [baptizō] since εἰς [eis] and ἐν [en] are really the same word in origin. In Acts 10:48 ἐν τῳ ὀνοματι Ἰησου Χριστου [en tōi onomati Iēsou Christou] occurs, but εἰς [eis] to ὀνομα [onoma] in 8:16 and 19:5. The use of ὀνομα [onoma] means in the name or with the authority of one as εἰς ὀνομα προφητου [eis onoma prophētou] (Matt. 10:41) as a prophet, in the name of a prophet. In the Acts the full name of the Trinity does not occur in baptism as in Matt. 28:19, but this does not show that it was not used. The name of Jesus Christ is the distinctive one in Christian baptism and really involves the Father and the Spirit. See on Matt. 28:19 for discussion of this point. “Luke does not give the form of words used in baptism by the Apostles, but merely states the fact that they baptized those who acknowledged Jesus as Messiah or as Lord” (Page). Unto the remission of your sins (εἰς ἀφεσιν των ἁμαρτιων ὑμων [eis aphesin tōn hamartiōn hūmōn]). This phrase is the subject of endless controversy as men look at it from the standpoint of sacramental or of evangelical theology. In themselves the words can express aim or purpose for that use of εἰς [eis] does exist as in 1 Cor. 2:7 εἰς δοξαν ἡμων [eis doxan hēmōn] (for our glory). But then another usage exists which is just as good Greek as the use of εἰς [eis] for aim or purpose. It is seen in Matt. 10:41 in three examples εἰς ὀνομα προφητου, δικαιου, μαθητου [eis onoma prophētou, dikaiou, mathētou] where it cannot be purpose or aim, but rather the basis or ground, on the basis of the name of prophet, righteous man, disciple, because one is, etc. It is seen again in Matt. 12:41 about the preaching of Jonah (εἰς το κηρυγμα Ἰωνα [eis to kērugma Iōna]). They repented because of (or at) the preaching of Jonah. The illustrations of both usages are numerous in the N. T. and the Koiné generally (Robertson, Grammar, p. 592). One will decide the use here according as he believes that baptism is essential to the remission of sins or not. My view is decidedly against the idea that Peter, Paul, or any one in the New Testament taught baptism as essential to the remission of sins or the means of securing such remission. So I understand Peter to be urging baptism on each of them who had already turned (repented) and for it to be done in the name of Jesus Christ on the basis of the forgiveness of sins which they had already received. The gift of the Holy Ghost (την δωρεαν του ἁγιου πνευματος [tēn dōrean tou hagiou pneumatos]). The gift consists (Acts 8:17) in the Holy Spirit (genitive of identification). [Robertson, A. T. (1933). Word Pictures in the New Testament (Ac 2:38). Broadman Press.]
I do not expect you to read this thoroughly or to consider A. T. Robertson's analysis or conclusions. But if you do read it and have questions, I can try to help you.
