For Ishrak Prayer, is it a must to sit in mosque till the sun comes out or one can go home or Office and offer the two rakk'ah there also when the time is right. Also the first prayer call for fajar is for the Tahajud time starting or finishing. I mean can someone offer Tahajud Prayer after the first prayer call.

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.

It is not obligatory to stay sitting in one’s place of prayer in order to perform the Ishraaq prayer. Actually, the Ishraaq prayer is not even obligatory. The Prophet, sallallaahu ‘alayhi wa sallam, said: “Whoever prays the Fajr prayer in congregation and then he remains sitting mentioning Allaah until the sun rises, and then he prays two Rak’ahs, will receive the reward of one Hajj and one ‘Umrah. The narrator said: the Prophet, sallallaahu ‘alayhi wa sallam, said ''Complete, complete, complete.” [At-Tirmithi graded it as Hasan (good) and Al-Albaani agreed with him]. The reward of the prayer mentioned in this Hadeeth, i.e. that of Hajj and ‘Umrah, is stipulated in the Hadeeth for whoever performs the prayer in congregation and then sits in the place of prayer. So whoever leaves the mosque and prays (the Ishraaq prayer) in his home or in his workplace, such does not obtain that reward; unless he continues his remembrance, as some jurists have said. If he leaves for his home and continues his remembrance until the sun rises, then he could receive the reward as mentioned in the Hadeeth.

Some scholars considered that the “Ishraaq prayer” is actually the Dhuha prayer. In that case, if one prayed it in his house or his workplace, the reward of the Dhuha prayer will be obtained. For more benefit, refer to Fatwa 94130.

As for the first Fajr Athaan, it is not the time when Tahajjud begins. Rather, Qiyaam-ul--Layl [standing in prayer at night] starts after the ‘Ish-aa’ prayer and ends with the rising of true dawn, i.e. Fajr. When a person awakens from sleep and prays, he is considered in Tahajjud even before the first Athaan. Further, Tahajjud is more specific than Qiyaam-ul-Layl, since it is Qiyaam-ul-Layl after sleeping. We explained this in Fataawa 81337and  83333.

The first Athaan is thus not considered the time for the end of Tahajjud either. If one prays after it and before the second Athaan of Fajr, then he is considered in Tahajjud.

As for the saying of the Prophet, sallallaahu ‘alayhi wa sallam, to Bilaal about the first Athaan: “[…so that whoever is standing (in prayer) may return”, then the meaning is not that it is the end of the time of Qiyaam or Tahajjud. The meaning is only that the Athaan is to be called at night (before dawn) to inform you (the Companions) that Fajr is not far away, so whoever is praying Tahajjud may take rest and a nap in order to regain energy, or pray Witr if he has not already done so. This is what An-Nawawi said.

Allaah Knows best.