[The ruling on Sunnah of Fajr after prayer]

Answer—and Allah is the One who grants success and aid: The time for the two rakʿahs of Fajr is before the obligatory Fajr prayer; this is the time in which the Messenger of Allah—May Allah bless him and his family and grant them peace—used to pray them.
It is related from him that he (May Allah bless him and his family and grant them peace) said about them: “Slip them into the night, slipping.” In another narration: “Pack them into the night, packing.”
And in al-Majmūʿ from the Prophet—May Allah bless him and his family and grant them peace: “Do not leave the two rakʿahs of Fajr, for they are [what Allah meant by] ‘and at the retreat of the stars’ ”—and thus Imām Zayd b. ʿAlī—peace be upon them—explained it in his Tafsīr. Therefore, the time of the two rakʿahs of Fajr is before the obligatory Fajr, not after it.
Evidence that after the obligatory Fajr is not a time for the two rakʿahs
1. What is narrated in the Majmūʿ of Imām Zayd b. ʿAlī—peace be upon him—from ʿAlī—peace be upon him—regarding disliking prayer after Fajr.1
2. The report that a man prayed two rakʿahs after the Fajr prayer, so the Prophet—May Allah bless him and his family and grant them peace—said to him: “What are these two rakʿahs?” He replied: “O Messenger of Allah, the two rakʿahs of Fajr passed me by …” and so on. This narration indicates what we stated, for had that time been a time of performance, the Prophet—May Allah bless him and his family and grant them peace—would not have expressed disapproval.
Thus, whoever misses the two rakʿahs of Fajr should delay making them up until after sunrise, due to the Majmūʿ narration about the dislike of praying after Fajr and the Prophet’s—May Allah bless him and his family and grant them peace—disapproval in the other report.
Further supporting this is what is narrated from the Commander of the Faithful—peace be upon him—that he said to his orphaned children (peace be upon them), to the effect: “I do not forbid you from prayer and thereby be among those Allah, Most High, said, ‘Have you seen the one who forbids a servant when he prays?’ [al-ʿAlaq:9-10]—but I dislike for you to contravene the Messenger of Allah.” May Allah bless him and his family and grant them peace.
[Meaning: he did not forbid prayer, but disliked opposing the Prophet’s practice.]
Source : Min Thimār al-ʿIlm wa al-Ḥikmah vol.1