Is it permissible for a woman to recite aloud during a silent prayer out of forgetfulness, and is it permissible for her to recite the supplications of bowing (rukūʿ) and prostration (sujūd) aloud?

The worshiper should not vocalize in a silent prayer nor silence in a vocal prayer. For the Sunnah is to vocalize in its proper place and to be silent in its proper place. And whoever vocalizes in a silent prayer, their prayer is valid, though their action is contrary to what is better (khilaf al-awla), and they should perform the prostration of forgetfulness (sujud al-sahw) after the salutation if they were forgetful. What is prescribed for supplication (du'a) and glorification (tasbih) in bowing (ruku') and prostration (sujud) is to say them silently.