Firstly: 

If a woman persists an immoral action and does not repent from it or give it up, even if the matter did not go as far as zina, such as if she had a relationship with this Christian man or anyone else, then it is not permissible for the husband to keep her, because that is a kind of cuckoldry (diyaathah), and cuckoldry is a major sin, because of the report narrated by al-Nasaa’i (2562) from ‘Abd-Allah ibn ‘Umar (may Allah be pleased with him) who said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “There are three at whom Allah, may He be glorified and exalted, will not look on the Day of Resurrection: the one who is defiant towards his parents, the woman who imitates men, and the cuckold.”

Classed as saheeh by al-Albaani in Saheeh al-Nasaa’i. 

The cuckold is the one who approves of evil conduct in his family. 

Shaykh al-Islam Ibn Taymiyah (may Allah be pleased with him) was asked about the one who came into his house and found a stranger with his wife, so he gave her her dues and divorced her by talaaq, then he went back and reconciled with her and heard that she had been found with a non-mahram man.  

He replied: In the hadeeth from the Prophet (blessings and peace of Allah be upon him) it says that when Allah, may He be glorified and exalted, created Paradise, He said: “By My glory and majesty, no miser, liar or cuckold will enter you.” The cuckold is the one who has no protective jealousy or pride. In al-Saheeh it is narrated that the Prophet (blessings and peace of Allah be upon him) said: “The believer has protective jealousy (gheerah) and Allah has protective jealousy, and the protective jealousy of Allah is that no slave should do that which is forbidden to him.” And Allah says (interpretation of the meaning): “The adulterer — fornicator marries not but an adulteress — fornicatress or a Mushrikah; and the adulteress –fornicatress, none marries her except an adulterer — fornicater or a Mushrik [and that means that the man who agrees to marry (have a sexual relation with) a Mushrikah (female polytheist, pagan or idolatress) or a prostitute, then surely, he is either an adulterer — fornicator, or a Mushrik (polytheist, pagan or idolater). And the woman who agrees to marry (have a sexual relation with) a Mushrik (polytheist, pagan or idolater) or an adulterer — fornicator, then she is either a prostitute or a Mushrikah (female polytheist, pagan, or idolatress)]. Such a thing is forbidden to the believers (of Islâmic Monotheism)” [al-Noor 24:3]. The correct scholarly opinion is that it is not permissible to marry a zaaniyah (a woman who commits fornication or adultery) until after she has repented. The same applies if the wife commits zina: the husband has no right to keep her in that case; rather he should leave her, otherwise he will be a cuckold (duyooth). End quote from Majmoo‘ al-Fataawa, 32/141

 Secondly: 

If the woman has repented, turned over a new leaf and is living a righteous life, and she has cut off all ties to that non-mahram man, then the husband may keep her, and perhaps Allah will reward him for treating her kindly and concealing her sin. 

We have stated that it is not permissible for him to keep her as his wife if she committed zina and has not repented from it sincerely, and we stated that if she repents and turns over a new leaf, then he may keep her and conceal her sin, if he has the patience to do that. What we have said about it being permissible to keep her if she repents is not obligatory for him, rather it is up to him. In all cases he may leave her, because zina on the part of the wife is extremely abhorrent and most people cannot forgive that. And if he divorces her, then he is not responsible for what she commits of sin, and if she apostatises from Islam, she alone is responsible for that, because she is accountable and of sound mind: if she does good deeds that it is in her favour and if she does bad deeds then it counts against her. 

We ask Allah to protect the Muslims from all trials, evils and turmoil. And Allah knows best.