What is the expiation for lying and altering the truth, then swearing upon the Mus'haf with an intention different from the apparent meaning due to necessity? And what is the ruling on a man teaching the Qur'an to a group of women?

Lying is forbidden (haram), but it is permissible to resort to tawriya (equivocation) and ma'areedh (allusive speech) to avoid it. These involve using words from which the listener understands something other than what the speaker intends. If this is not possible, lying is permissible in case of necessity, or to achieve a religiously sanctioned benefit, or to ward off harm. It may even become obligatory if it is for the attainment of an obligatory matter; for example, if a Muslim hides from an oppressor, then lying to conceal him becomes obligatory.

Therefore, if the oath is out of necessity and the one administering the oath has no right, there is no sin upon the swearer as long as his intention is correct, with the possibility of offering an expiation for the oath (kaffarat yamin) as a precaution. As for a man teaching women, there is no harm in it if it adheres to Islamic guidelines.