Is it permissible for me not to mention what happened to me in the security investigation, and to suffice with saying that I fell, in order to avoid exposing my wife'sستر (cover/privacy) in an incident that led to my hand being broken?

Lying is forbidden, and warnings have been issued against it, clarifying its evil consequences and that it contradicts faith. The sin is compounded by swearing a false oath, as it is a "ghamus" oath (an oath that immerses the swearer in sin), which immerses its taker in Hellfire or in sin. Despite the gravity of lying and swearing by it and the obligation to avoid it, scholars have mentioned that if a preponderant benefit necessitates it, there is no harm in it. Thus, if there is a recognized benefit and a certain harm is avoided, and it does not involve harming others, then lying and swearing falsely are permissible if the situation demands it.