Is it permissible to lie by saying "I have prayed" during menstruation to avoid embarrassment, with the intention of performing another prayer later, and does this apply only to young people or does it include women as well?

Deliberate lying is forbidden (haram), but it can be avoided by using tawriya (equivocation) and ma'arid (ambiguous expressions). These are general linguistic formulations from which the listener understands something other than what the speaker intends. In ma'arid, there is room to avoid lying. There is no harm in saying, "I prayed," intending a missed prayer or invoking blessings upon the Prophet (peace be upon him). There is no difference in this regard between women and men.