Your Question: Who is permitted to preach a homily at Mass?

The preaching of a homily at Mass is reserved for ordained persons, such as the priest or deacon. Laypersons are permitted to preach in certain circumstances in a church, but it should not take the place of the homily at Mass. So, for example, if someone is giving a “witness talk” of some kind at Mass, which can be beneficial in certain circumstances, it should not be give in the place of a homily, but at some other time in the Mass.

For more information on this topic, see
http://www.jimmyakin.org/2004/05/lay_homilists.html