What is the ruling on supplicating aloud with raised hands between the two rak'ahs of Fajr prayer, singling out Fajr prayer for this supplication over others, and have any hadiths been narrated concerning this?

The question is unclear, but if what is meant is the supplication (du'a) between the Adhan and Iqama, then the Sunnah has established the desirability of making supplication during this time, based on the Prophet's (peace be upon him) saying: "The supplication between the Adhan and the Iqama is not rejected." However, no specific supplication has been authentically designated for this time, and everything reported in that regard is weak. Designating a specific supplication to be recited aloud and consistently is not from the Sunnah.

If what is meant is the Qunut supplication in the Fajr prayer, scholars have differed regarding its permissibility. Some have deemed it desirable, citing the Hadith of Anas: "The Messenger of Allah (peace be upon him) continued to perform Qunut until he departed from this world." However, this Hadith is weak. Those who argue against its permissibility cite the Hadith of Sa'd ibn Tariq Al-Ashja'i, which indicates that the Companions did not perform Qunut in the Fajr prayer. The evidence of those who argue against the desirability of Qunut is stronger.

Nevertheless, this issue is one of ijtihad (independent reasoning) and should not be a cause for strictness. Scholars have agreed on the validity of the prayer of one who performs Qunut and one who does not. If the Imam performs Qunut, the worshiper behind him should follow.