How can the statement of Sheikh Al-Islam Ibn Taymiyyah that a wrongdoing can be tolerated in two cases (averting that which is worse than it, and achieving that which is more beneficial than abandoning it) be reconciled with the saying: "The end does not justify the means"?

There is no contradiction between the words of Shaykh al-Islam Ibn Taymiyyah regarding enduring a minor evil to avert a greater one, and the rule that the end does not justify the means. This is because rules have exceptions. The rule "the end does not justify forbidden means" applies to someone who steals or commits adultery for a praiseworthy goal, in which case neither theft nor adultery becomes permissible. However, a praiseworthy end may justify a forbidden means if it is a greater benefit that cannot be achieved except through it, such as permissible lying to avert a greater evil, or backbiting if it serves a legitimate Sharia purpose that cannot be achieved otherwise, like warning Muslims, or mentioning the openly immoral person by what they have openly done. Ibn al-Jawzi says: "Every praiseworthy objective that cannot be attained except through lying is permissible if the objective itself is permissible, and if it is obligatory, then lying becomes obligatory." Ibn al-Qayyim says: "It is permissible for a person to lie about himself or about others, as long as it does not involve harm to that other person, if the lie leads to the person's right." Similarly, a minor evil may be tolerated to avert a greater evil, or to achieve a greater benefit. And a good deed may be abandoned if it causes the loss of something better, or if it necessitates an evil whose harm outweighs its benefit.