Answer –

It is a Sunnah to stay awake after the Fajr Salaah engaging in the Thikar  of Allah until the sun has risen, and to then pray 2 or 4  Rakaats  of Salatul Ishraq.

The time for performing Salah Ishraq commences approximately twenty minutes after sunrise when the sun is above the horizon approximately to the length of a spear

قَالَ رَسُولُ اللَّهِ صلى الله عليه وسلم ‏ ‏ لأَنْ أَقْعُدَ مَعَ قَوْمٍ يَذْكُرُونَ اللَّهَ تَعَالَى مِنْ صَلاَةِ الْغَدَاةِ حَتَّى تَطْلُعَ الشَّمْسُ أَحَبُّ إِلَىَّ مِنْ أَنْ أُعْتِقَ أَرْبَعَةً مِنْ وَلَدِ إِسْمَاعِيلَ

Nabi Sallallahu Alaihi Wasallam said, “That I sit in the company of the people who remember Allah the Exalted from Fajr Salaah till the sun rises is dearer to me than that I emancipate four slaves from the children of Ismaeel.  [Abu Dawud Book 26, Hadith 27]

Imam Tirmidhi (رضي الله عنه) narrates:

قَالَ رَسُولُ اللَّهِ صلى الله عليه وسلم ‏ مَنْ صَلَّى الْغَدَاةَ فِي جَمَاعَةٍ ثُمَّ قَعَدَ يَذْكُرُ اللَّهَ حَتَّى تَطْلُعَ الشَّمْسُ ثُمَّ صَلَّى رَكْعَتَيْنِ كَانَتْ لَهُ كَأَجْرِ حَجَّةٍ وَعُمْرَةٍ

Nabi Sallallahu Alaihi Wasallam said, “Whoever performs Fajr in congregation, then sits remembering Allah until the sun has risen, then he performs  two rak’ah, then for him is the reward like that of a Hajj and Umrah.” [Tirmidhi] The same reward will apply for a lady performing Salaah in her home.

We are able to gauge from these narrations that  Nabi (Sallallaahu Alayhi Wasallam) would not sleep after Fajr Salaat, but would remain in the Dhikr of Allah, the tilaawat of the Quraan, etc. until Ishraaq, Nabi Sallallahu Alaihi Wasallam would  perform Ishraq Salaah  after which he would engage in other chores. After Zawaal, Nabi Sallallahu Alaihi Wasallam would have a siesta.

It is makrooh (disliked) for one to sleep after Fajr till sunrise unless one has an excuse. Furthermore, it is not makrooh for one to sleep after sunrise.

ويكره النوم قبل العشاء والكلام المباح بعدها وبعد طلوع الفجر إلى أدائه ثم لا بأس بمشيه لحاجته وقيل يكره إلى طلوع ذكاء وقيل إلى ارتفاعها فيض (الدر المختار 1/ 381)

ويكره النوم في أول النهار وفيما بين المغرب والعشاء (هندية 376/5)

And Allah Ta’ala Knows Best

Yusuf Moosagie