There is a punishment for adultery, but it is not a husband’s duty or right to carry out. And the punishment for a man who commits adultery is equal to a woman’s. In either case, a trial, with specific standards of evidence, must take place.
Qur-an 24.2 can be translated as
“Those who commit unlawful sex- whether male or female- flog them with a hundred lashes”
So the punishment for males and females is equal. Since we all know that there is a double-standard when it comes to men’s and women’s sexual activity and its effects on their reputations, it should be a relief to know that Allah Commanded that
“those who accuse honorable women, but do not produce four witnesses, flog them with eighty lashes, and do not admit their testimony ever after” (Qur-an 24.4)
This applies even to husbands, as Qur-an 24.6-9 illustrate.
“As for those who accuse their wives, and have no witnesses except themselves: the testimony of such a one is that he testify, swearing by Allah four times that he is truthful, and a fifth time, that the curse of Allah be on him if he is lying. And the punishment shall be averted from the woman if she were to testify, swearing by Allah four times that the man was lying, and a fifth time that the wrath of Allah be upon her if the man be truthful.”
The calling of witnesses implies an audience. As such, it is not the husband’s prerogative or “right” to punish his wife. Rather, a public trial must take place that has witnesses and a chance for the accused to defend themselves.
Now, can a woman accuse her husband of adultery? Qur-an 24.2 makes it clear that men are also liable, so obviously anybody can accuse anybody of fornication or adultery, but there have to be four witnesses, except when the accuser is a spouse.