What are the situations in which it is impermissible to raise one's hands in supplication, and is it permissible to raise one's hands in supplication during the Friday prayer secretly during the sitting for rest, and to say "Ameen" behind the Imam at the end of the sermon before the establishment of the prayer?

Raising the hands while supplicating is not legislated in specific instances: during prayer, and for the Imam during the Friday sermon, except in the case of seeking rain (Istisqa'). Other than these, raising the hands is legislated, because the default for supplication is raising the hands, due to the Prophet's (peace be upon him) saying: "Indeed, your Lord, the Blessed, the Most High, is Modest and Generous. He is shy to return the hands of His servant empty when he raises them to Him." Nothing deviates from this principle except what the Prophet (peace be upon him) consistently supplicated in without it being reported that he raised his hands. As for raising the hands for supplication during the preacher's sitting between the two sermons, there is no harm in it, according to the preponderant opinion. As for the short sitting for rest (jalsat al-istiraha) during prayers, there is no supplication in it due to its brevity.