Virtues of Staying in the Mosque After Fajr
Question

I have friends who stay in the masjid after Fajr prayer to mention Allah individually. Others sit in a group and recite the Quran together in the same masjid. What is your opinion about this? Is it better to read the Adhkar of day and night in the masjid or at home?
Praise be to Allah.Virtues of staying in the mosque after Fajr
It is recommended to stay in the mosque after Fajr prayer until the sun rises, because this is what the Prophet (peace and blessings of Allah be upon him) and his companions did, and because there is great reward in that. Muslim (670) narrated from Jabir ibn Samurah (may Allah be pleased with him) that when the Prophet (peace and blessings of Allah be upon him) had prayed Fajr, he would sit in his prayer place until the sun had risen properly. 
Muslim also narrated (670) that Sammak ibn Harb (may Allah have mercy on him) said: I said to Jabir ibn Samurah: Did you sit with the Messenger of Allah (peace and blessings of Allah be upon him)? He said: Yes, frequently. He would not get up from the place in which he had prayed Fajr until the sun had risen, and when the sun had risen he would get up. They used to chat and talk about matters of the Jahiliyyah (the pre-Islamic period), and they would laugh but he smiled. 
At-Tirmidhi narrated that Anas ibn Malik (may Allah be pleased with him) said: The Messenger of Allah (peace and blessings of Allah be upon him) said: “Whoever prays Fajr in congregation then sits remembering Allah until the sun rises, then prays two rak’ahs, will have a reward like that of Hajj and ‘Umrah .” He said: The Messenger of Allah (peace and blessings of Allah be upon him) said: “In full, in full, in full.”  
This Hadith was classed as sound by Al-Albani in Sahih At-Tirmidhi. 
This points to the virtue of sitting in the mosque after Fajr prayer, and the virtue of praying two Rak`ahs when the sun has risen fully. 
In general, if a person stays in the mosque to remember Allah (Dhikr) and worship Him, or waiting for the prayer, all of these are good deeds and acts of worship that bring one closer to Allah. Al-Bukhari (445) and Muslim (649) narrated from Abu Hurayrah (may Allah be pleased with him) that the Messenger of Allah (peace and blessings of Allah be upon him) said: “The angels send blessings upon one of you so long as he is in his prayer-place where he offered his prayer, so long as he does not break his wudu; they say: ‘O Allah forgive him, O Allah have mercy on him.’” 
Based on this, your brothers have done well by staying in the mosque to remember Allah, and we ask Allah to reward them. 
Reading Quran in unison
Reading the Quran together in unison is not prescribed, because it is not mentioned in the Sunnah. And if it is done in a loud voice that disturbs those who are remembering Allah and sitting there, then it is even more disliked, because of the general meaning of the words of the Prophet (peace and blessings of Allah be upon him): “When one of you stands in prayer, he is conversing with his Lord, so let one of you know what he is saying to his Lord and do not raise your voices above one another in reciting when praying.” (Narrated by Ahmad, 4928 and classed as authentic by Shu’ayb Al-Arna’ut in Tahqiq Al-Musnad). 
It says in Fatawa Al-Lajnah Ad-Da’imah (4/112): What is the ruling on reciting Quran together in the mosque? 
Answer: 
“The question is somewhat vague. If what is meant is that they recite together in unison, pausing and stopping at the same time, this is not prescribed, and at the very least it is makruh, because there is no report from the Prophet (peace and blessings of Allah be upon him) or from the Companions that they did that. But if that is done for the purpose of teaching, then we hope that there is nothing wrong with it. But if what is meant is that they gather to read Quran in order to memorize it or learn it, and one of them reads whilst the others listen, or each of them reads to himself without raising his voice or reading in unison with others, then that is prescribed, because it is proven that the Messenger of Allah (peace and blessings of Allah be upon him) said: “No people gather in one of the houses of Allah, reciting the Book of Allah and studying it together, but sakinah (tranquility) descends upon them and the angels surround them and mercy covers them, and Allah remembers them to those who are with Him.” (Narrated by Muslim)” 
It also says (4/39):  
|”Question: 
Is it permissible to recite Surat Ya-Sin out loud in the mosque or not? 
Answer: 
It is not permissible for anyone to raise his voice when reciting Qur’an in the mosque, whether it is Soorat Ya-Seen or any other surah of the Quran, whether in prayer or otherwise, because it is proven that the Prophet (peace and blessings of Allah be upon him) came out to the people when they were praying and reciting Quran out loud, and he said: “O people, each of you is conversing with his Lord, so do not raise your voices above one another in reciting.” And because by doing that they distract and disturb one another.” 
These readers should be advised and called in a gentle manner. 
And Allah knows best.