Who could baptize, and was baptism by heretics valid?

Ignatius of Antioch d. c. 110 ADLetter to the Smyrnaeans 8.2

It is not lawful without the bishop either to baptize or to hold the love feast; but whatever he approves, this is pleasing to God too, so that everything you do may be secure and valid.

The same passage first calls the Church "catholic".

Tertullian c. 155–c. 220 ADOn Baptism 17.1–2

The right of giving it belongs to the high priest, who is the bishop; then to presbyters and deacons, but not without the bishop's authority, for the honor of the Church, which when kept keeps the peace. Otherwise even laymen have the right, for what is received equally can be given equally. ... But all the more must laymen keep humility and restraint, so as not to take on themselves the bishop's office.

In an emergency any Christian may baptize. Tertullian goes on to forbid women to baptize (17.4).

Council of Carthage (Caecilius of Bilta)Opinions of the Bishops 1

I know one baptism, in the Church alone, and none outside the Church. Here there will be one, where hope is true and faith certain. For so it is written: one faith, one hope, one baptism (Ephesians 4:5); not among heretics, where hope is none and faith false.

The first of 87 bishops who voted at Cyprian's council that converts from heretical sects must be baptized.

Stephen of Rome (quoted by Cyprian) d. 257 ADLetter, quoted in Cyprian, Letter 74 74.1

If any come to you from any heresy whatever, let nothing be changed except what has been handed down: let hands be laid on them for repentance, since the heretics themselves do not baptize those who come to them from one another, but only give them communion.

Cyprian quotes the bishop of Rome in order to refute him. Each side claimed tradition.

Anonymous fl. c. 150 ADOn Rebaptism 10

Therefore help should be given to those who repent and are corrected by the teaching of the truth and by their own faith, which was afterwards set right and their heart purified, only by the baptism of the Spirit, that is, by the laying on of the bishop's hand and the giving of the Holy Spirit.

The anonymous author, answering Cyprian in the rebaptism controversy, defends the old custom: a baptism in the name of Jesus, though given in heresy, is not repeated; what the returning heretic still lacks is the Spirit, given by the bishop's hand. The work circulated with Cyprian's writings and is ascribed to him in the manuscripts.

Firmilian of Caesarea d. c. 269 ADLetter to Cyprian (Letter 75 among Cyprian's letters) 75.8

Since Stephen and those who agree with him claim that forgiveness of sins and a second birth can proceed through heretics' baptism, among whom even they confess the Holy Spirit is not, let them consider and understand that a spiritual birth cannot be without the Spirit. ... What then of this: when we see that Paul baptized his disciples again with a spiritual baptism after John's baptism, do we hesitate to baptize those who come to the Church from heresy after their unlawful and profane dipping?

Firmilian, bishop of Caesarea in Cappadocia, writes to Cyprian in full agreement with the African council of 256: heretical baptism is no baptism, so those who come from heresy are to be baptized. His letter survives as Letter 75 in Cyprian's collection.

Augustine of Hippo 354–430 ADOn Baptism, against the Donatists 4.17.24

There is no salvation outside the Church. Who denies it? And so whatever belongs to the Church has no value for salvation outside the Church. But it is one thing not to have, another not to have to profit. Whoever does not have it must be baptized to have it; whoever has it without profit must be corrected so as to have it with profit. Nor is the water adulterous in the baptism of heretics.

Augustine answers the Donatists, who appealed to Cyprian. Baptism belongs to Christ, not to the minister, so it is not repeated.