Is it permissible to pray two rak'ahs of Ishraq in the mosque, then pray four rak'ahs of Duha after two hours at home?

The Ishraq prayer is the first part of the Duha prayer, and it is named so because it is performed at the very beginning of the Duha time, after the sun has risen to the height of a spear and the time of dislikability (for prayer) has passed. It is also called the Ishraq prayer.

Its virtue is evidenced by the hadith narrated by Anas (may Allah be pleased with him), in which the Prophet (peace and blessings be upon him) said: "Whoever prays Fajr in congregation, then sits remembering Allah until the sun rises, then prays two rak'ahs, he will have a reward like that of a complete, complete, complete Hajj and 'Umrah."

Consecutiveness is not required for the Ishraq prayer immediately after the Fajr prayer. A Muslim may pray Ishraq and then pray afterwards until the time of dislikability.