Objection 1. It would seem that solemn penance should not be imposed on women. Because, when this penance is imposed on a man, he has to cut his hair off. But this becomes not a woman, according to 1 Corinthians 11:15. Therefore she should not do solemn penance.
Objection 2. It also seems that it ought to be imposed on clerics. For it is enjoined on account of a grievous crime. Now the same sin is more grievous in a cleric than in a layman. Therefore it ought to be imposed on a cleric more than on a layman.
Objection 3. It also seems that it can be imposed by any priest. Because to absolve in the tribunal of Penance belongs to one who has the keys. Now an ordinary priest has the keys. Therefore he can administer this penance.
I answer that, Every solemn penance is public, but not vice versa. For solemn penance is done as follows: "On the first day of Lent, these penitents clothed in sackcloth, with bare feet, their faces to the ground, and their hair shorn away, accompanied by their priests, present themselves to the bishop of the city at the door of the church. Having brought them into the church the bishop with all his clergy recites the seven penitential psalms, and then imposes his hand on them, sprinkles them with holy water, puts ashes on their heads, covers their shoulders with a hairshirt, and sorrowfully announces to them that as Adam was expelled from paradise, so are they expelled from the church. He then orders the ministers to put them out of the church, and the clergy follow reciting the responsory: 'In the sweat of thy brow,' etc. Every year on the day of our Lord's Supper they are brought back into the church by their priests, and there shall they be until the octave day of Easter, without however being admitted to Communion or to the kiss of peace. This shall be done every year as long as entrance into the church is forbidden them. The final reconciliation is reserved to the bishop, who alone can impose solemn penance" [Cap. lxiv, dist. 50].
This penance can be imposed on men and women; but not on clerics, for fear of scandal. Nor ought such a penance to be imposed except for a crime which has disturbed the whole of the city.
On the other hand public but not solemn penance is that which is done in the presence of the Church, but without the foregoing solemnity, such as a pilgrimage throughout the world with a staff. A penance of this kind can be repeated, and can be imposed by a mere priest, even on a cleric. Sometimes however a solemn penance is taken to signify a public one: so that authorities speak of solemn penance in different senses.
Reply to Objection 1. The woman's hair is a sign of her subjection, a man's is not. Hence it is not proper for a woman to put aside her hair when doing penance, as it is for a man.
Reply to Objection 2. Although in the same kind of sin, a cleric offends more grievously than a layman, yet a solemn penance is not imposed on him, lest his orders should be an object of contempt. Thus deference is given not to the person but to his orders.
Reply to Objection 3. Grave sins need great care in their cure. Hence the imposition of a solemn penance, which is only applied for the most grievous sins, is reserved to the bishop.
Source: Summa Theologica (New Advent)