Praise be to Allah.
Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) said :
On the contrary, he should recite quietly, because when a person who joined the prayer late stands up to make up what he missed, he is then praying alone, so at that point he is subject to the rulings concerning one who prays alone. During the part of the (congregational) prayer he caught up with, he was subject to the rulings concerning one who prays in congregation behind an imaam. So he should do sujood (sajdat al-sahw – prostration of forgetfulness) if he forgets something in the part of the prayer that he is making up. Since this is the case, then the one who joined the prayer late should recite aloud where the person who prays individually should recite aloud. The scholars whose view is that the person who prays individually should recite aloud in Maghrib, ‘Ishaa’ and Fajr say that he should recite aloud if he is making up the first two rak’ahs. The scholars whose view is that the person who prays individually should not recite aloud say that the person who joined the prayer late should not recite aloud in this case. Nobody prays Jumu’ah individually, so no one can imagine that a person making up part of it individually has to recite aloud. The person who joined the prayer late is like the person who is praying individually, so he should not recite aloud. But the person who joins the Jumu’ah prayer late is considered to have prayed Jumu’ah in essence, because he followed the imaam, and different conditions apply to the one who follows than to the one who leads, with regard to numbers and so on. But the Sunnah states that a person who catches up with one rak’ah has caught up with the prayer, so he has caught up with Jumu’ah, just as a person who does one rak’ah of ‘Asr before the sun sets, or one rak’ah of Fajr before the sun rises, is considered to have caught up with the prayer and done it on time, even if the rest of his prayer is done outside of the prescribed time. And Allaah knows best.