On the Homily and Who can give it

Catechism of the Catholic Church #1142

But "the members do not all have the same function." Certain members are called by God, in and through the Church, to a special service of the community. These servants are chosen and consecrated by the sacrament of Holy Orders, by which the Holy Spirit enables them to act in the person of Christ the head, for the service of all the members of the Church. The ordained minister is, as it were, an "icon" of Christ the priest. Since it is in the Eucharist that the sacrament of the Church is made fully visible, it is in his presiding at the Eucharist that the bishop's ministry is most evident, as well as, in communion with him, the ministry of priests and deacons.

Code of Canon Law

Can. 767 §1. Among the forms of preaching, the homily, which is part of the liturgy itself and is reserved to a priest or deacon, is preeminent; in the homily the mysteries of faith and the norms of Christian life are to be explained from the sacred text during the course of the liturgical year.

§4. It is for the pastor or rector of a church to take care that these prescripts are observed conscientiously.

General Instruction on the Roman Missal

66. The homily should ordinarily be given by the priest celebrant himself. He may entrust it to a concelebrating priest or occasionally, according to circumstances, to the deacon, but never to a lay person. In particular cases and for a just cause, the homily may even be given by a Bishop or a priest who is present at the celebration but cannot concelebrate.

Instruction Redemptis Sacramentum, Congregation for Divine Worship and the Discipline of the Sacraments (2004)

(RS 66) The prohibition of the admission of laypersons to preach within the Mass applies also to seminarians, students of theological disciplines, and those who have assumed the function of those known as “pastoral assistants” ; nor is there to be any exception for any other kind of layperson, or group, or community, or association.

On Certain Questions Regarding the Collaboration of the non-ordained faithful in the Sacred Ministry of the Priest (1997)

The homily, therefore, during the celebration of the Holy Eucharist, must be reserved to the sacred minister, Priest or Deacon(69) to the exclusion of the non-ordained faithful, even if these should have responsibilities as "pastoral assistants" or catechists in whatever type of community or group. This exclusion is not based on the preaching ability of sacred ministers nor their theological preparation, but on that function which is reserved to them in virtue of having received the Sacrament of Holy Orders. For the same reason the diocesan Bishop cannot validly dispense from the canonical norm since this is not merely a disciplinary law but one which touches upon the closely connected functions of teaching and sanctifying.

For the same reason, the practice, on some occasions, of entrusting the preaching of the homily to seminarians or theology students who are not clerics is not permitted. Indeed, the homily should not be regarded as a training for some future ministry.

All previous norms which may have admitted the non-ordained faithful to preaching the homily during the Holy Eucharist are to be considered abrogated by canon 767, § 1.

§ 2. A form of instruction designed to promote a greater understanding of the liturgy, including personal testimonies, or the celebration of eucharistic liturgies on special occasions (e.g. day of the Seminary, day of the sick etc.) is lawful, of in harmony with liturgical norms, should such be considered objectively opportune as a means of explicating the regular homily preached by the celebrant priest. Nonetheless, these testimonies or explanations may not be such so as to assume a character which could be confused with the homily.