Praise be to Allah.
If a person misses Fajr prayer in congregation and did not catch up with it, then he should start with the Sunnah prayer and offer that first, then pray Fajr. When the Prophet (sa) slept one night and missed Fajr prayer whilst travelling, and nothing woke him up except the heat of the sun, he offered the Sunnah prayer then he prayed Fajr, and he gave instructions that the adhaan and iqaamah should be given as was usual. So if a person misses Fajr prayer and does not wake up until after the people have prayed, or he comes to the mosque and finds that the people have already prayed, then he should offered the Sunnah prayer of two rak‘ahs, then pray Fajr. This is the Sunnah.
But if he delays it until after the prayer or after the sun has risen, there is nothing wrong with that. But it is preferable to start with it, as the Prophet (blessings and peace of Allah be upon him) did when he missed the prayer on one of his journeys.
If he comes to the congregation and finds them already praying and he prays with them without having offered the Sunnah prayer first, he may offer it after the (obligatory) prayer, to make it up, or he may offer it after the sun has risen and become high, which is preferable. Both options are mentioned in the hadeeth, and he has the choice, in sha Allah: he may offer it after Fajr, in the mosque or at home, or if he wishes he may offer it after the sun has risen and become high. The matter is broad in scope, praise be to Allah, and both are mentioned in hadeeths from the Prophet (blessings and peace of Allah be upon him). End quote.
Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allah have mercy on him)