Objection 1. It would seem that a divine person is sent only by the one whence He proceeds eternally. For as Augustine says (De Trin. iv), "The Father is sent by no one because He is from no one." Therefore if a divine person is sent by another, He must be from that other.
Objection 2. Further, the sender has authority over the one sent. But there can be no authority as regards a divine person except from origin. Therefore the divine person sent must proceed from the one sending.
Objection 3. Further, if a divine person can be sent by one whence He does not proceed, then the Holy Ghost may be given by a man, although He proceeds not from him; which is contrary to what Augustine says (De Trin. xv). Therefore the divine person is sent only by the one whence He proceeds.
On the contrary, The Son is sent by the Holy Ghost, according to Isaiah 48:16, "Now the Lord God hath sent Me and His Spirit." But the Son is not from the Holy Ghost. Therefore a divine person is sent by one from Whom He does not proceed.
I answer that, There are different opinions on this point. Some say that the divine person is sent only by the one whence He proceeds eternally; and so, when it is said that the Son of God is sent by the Holy Ghost, this is to be explained as regards His human nature, by reason of which He was sent to preach by the Holy Ghost. Augustine, however, says (De Trin. ii, 5) that the Son is sent by Himself, and by the Holy Ghost; and the Holy Ghost is sent by Himself, and by the Son; so that to be sent in God does not apply to each person, but only to the person proceeding from another, whereas to send belongs to each person.
There is some truth in both of these opinions; because when a person is described as being sent, the person Himself existing from another is designated, with the visible or invisible effect, applicable to the mission of the divine person. Thus if the sender be designated as the principle of the person sent, in this sense not each person sends, but that person only Who is the principle of that person who is sent; and thus the Son is sent only by the Father; and the Holy Ghost by the Father and the Son. If, however, the person sending is understood as the principle of the effect implied in the mission, in that sense the whole Trinity sends the person sent. This reason does not prove that a man can send the Holy Ghost, forasmuch as man cannot cause the effect of grace.
The answers to the objections appear from the above.
Source: Summa Theologica (New Advent)