How can a husband forgive his adulterous wife and forget her deed, especially if he knows the adulterer and fears scandal and people's taunts, and in this case, is forgiveness still better, or is it permissible to cleanse the shame, which may even lead to murder?

If a wife commits adultery and then repents, it is best for her husband to forgive her and conceal her sin. It is not permissible for him to kill her due to the resulting corruption, the spread of scandal, and the act of murder. As for some people slandering Muslims' reputations, they must either produce clear evidence or have the Hadd punishment for false accusation applied to them. The ruler must punish such individuals to make them an example for others.