Objection 1. It would seem that it is fitting also that the Father should be sent. For being sent means that the divine person is given. But the Father gives Himself since He can only be possessed by His giving Himself. Therefore it can be said that the Father sends Himself.
Objection 2. Further, the divine person is sent according to the indwelling of grace. But by grace the whole Trinity dwells in us according to John 14:23: "We will come to him and make Our abode with him." Therefore each one of the divine persons is sent.
Objection 3. Further, whatever belongs to one person, belongs to them all, except the notions and persons. But mission does not signify any person; nor even a notion, since there are only five notions, as stated above (Question 32, Article 3). Therefore every divine person can be sent.
On the contrary, Augustine says (De Trin. ii, 3), "The Father alone is never described as being sent."
I answer that, The very idea of mission means procession from another, and in God it means procession according to origin, as above expounded. Hence, as the Father is not from another, in no way is it fitting for Him to be sent; but this can only belong to the Son and to the Holy Ghost, to Whom it belongs to be from another.
Reply to Objection 1. In the sense of "giving" as a free bestowal of something, the Father gives Himself, as freely bestowing Himself to be enjoyed by the creature. But as implying the authority of the giver as regards what is given, "to be given" only applies in God to the Person Who is from another; and the same as regards "being sent."
Reply to Objection 2. Although the effect of grace is also from the Father, Who dwells in us by grace, just as the Son and the Holy Ghost, still He is not described as being sent, for He is not from another. Thus Augustine says (De Trin. iv, 20) that "The Father, when known by anyone in time, is not said to be sent; for there is no one whence He is, or from whom He proceeds."
Reply to Objection 3. Mission, meaning procession from the sender, includes the signification of a notion, not of a special notion, but in general; thus "to be from another" is common to two of the notions.
Source: Summa Theologica (New Advent)