What is the ruling on swearing by Allah falsely to avoid annoying someone or harming their psyche, and what is its expiation, and is it permissible to give the expiation to a needy sister?

Lying is informing خلاف الواقع (contrary to reality). Swearing an oath upon it is an Al-Yamin Al-Ghamus (a false oath) that immerses its taker in Hellfire and is considered among the major sins. Swearing an oath upon a lie is forbidden and entails two corruptions: lying in itself and violating the عظَمة الله (greatness of Allah). If no significant harm results from telling the truth, then you must repent from lying and swearing an oath upon it. However, if harm would result, then you should have avoided lying by being truthful. If truthfulness was impossible, then lying was permissible without an oath. If an oath became necessary to avert harm, then it was also permissible. The majority of scholars believe that Al-Yamin Al-Ghamus has no expiation except repentance and seeking forgiveness, while Al-Shafi'i holds that expiation is obligatory. The more cautious approach is to combine both. It is permissible to give the expiation to your sister if she is among the poor and needy.