Assalamualaikum ustaz. When the time of Isyak arrives, sometimes in mosques, religious class or sermon was ongoing. Hence, is it permissible for the isyak adhan to be called after it is over, even when the time for isyak has begun for quite some time?
Waalaikumussalam wrt. wbt.,
Alhamdulillah, praise and gratitude to Allah SWT for His countless blessings for us all. Praise and salutations to our beloved Prophet Muhammad PBUH, his family, companions, and all those who follow his footsteps until the Final day.
Basically, Islamic teachings or classes held in mosques and attended by the congregants are the practise of previous salaf al-soleh. Rasullullah PBUH and his companions as well as the tabi’in, the four imams and renowned scholars have long made the mosques an education hub and this is accepted by the Muslim ummah.
We are certain that a mosque that has a full schedule of classes and sermons are included to mean with the meaning of mosques that are enlivened. So true is the statement of Allah SWT:
إِنَّمَا يَعْمُرُ مَسَاجِدَ اللَّـهِ مَنْ آمَنَ بِاللَّـهِ وَالْيَوْمِ الْآخِرِ وَأَقَامَ الصَّلَاةَ وَآتَى الزَّكَاةَ وَلَمْ يَخْشَ إِلَّا اللَّـهَ ۖ فَعَسَىٰ أُولَـٰئِكَ أَن يَكُونُوا مِنَ الْمُهْتَدِينَ
“The mosques of Allah are only to be maintained by those who believe in Allah and the Last Day and establish prayer and give zakah and do not fear except Allah, for it is expected that those will be of the [rightly] guided.” 
Abu Hayyan said this verse is a commandment for believers to enliven the mosque, by cleaning, honouring, worshipping, dhikr and holding religious classes. Indeed, a mosque is one of the gardens of paradise for believers in this world. 
While in a hadith narrated from Abu Hurairah RA, the Prophet PBUH said:
وَمَنْ سَلَكَ طَرِيقًا يَلْتَمِسُ فِيهِ عِلْمًا سَهَّلَ اللَّهُ لَهُ بِهِ طَرِيقًا إلَى الْجَنَّةِ، وَمَا اجْتَمَعَ قَوْمٌ فِي بَيْتٍ مِنْ بُيُوتِ اللَّهِ يَتْلُونَ كِتَابَ اللَّهِ، وَيَتَدَارَسُونَهُ فِيمَا بَيْنَهُمْ؛ إلَّا نَزَلَتْ عَلَيْهِمْ السَّكِينَةُ، وَغَشِيَتْهُمْ الرَّحْمَةُ، وَ حَفَّتهُمُ المَلاَئِكَة، وَذَكَرَهُمْ اللَّهُ فِيمَنْ عِنْدَهُ،
“And whoever follows a path to seek knowledge therein, Allah will make easy for him a path to Paradise. No people gather together in one of the Houses of Allah, reciting the Book of Allah and studying it among themselves, except that sakeenah (tranquility) descends upon them, and mercy envelops them, and the angels surround them, and Allah mentions them amongst those who are with Him.” 
According to Imam al-Nawawi, it is said that what is meant by Sakinah here is blessings. This is chosen by al-Qadhi ‘Iyadh but it is dhaif (weak) because it has been combined with Sakinah. It is also said that (what is meant with sakinah) is thama’ninah and honour. This is better and there are also evidences and guidance of the advantages of gathering to recite al-Quran in the mosque. 
Thus, according to this matter, we hope that may Allah make us from amongst those who love the mosque and knowledge ceremonies. Hence, we’ll be eligible to be from those who attain Allah’s paradise as stated in the above hadith.
In answering the above question, adhan can still be broadcasted even after the time for isyak has started for a long time. The author of the book al-Fiqh al-Manhaji stated: “And the adhan is sunnah for a prayer even after the time for the adhan has expired.” 
However, we suggest that it is best if the sermon is paused to give way for the Isyak’s adhan to be called out. Afterwards, continue the rest of the sermon. The reason is it serves as a reminder for those outside of the mosque and at home that the time for prayer has begun. While the time after the adhan can be sued for people to gather to pray before iqamat. According to a hadith of Jabir bin Abdillah RA regarding the Isyak prayer of Rasullullah PBUH:
إِذَا رَآهُمُ اجْتَمَعُوا عَجَّلَ، وَإِذَا رَآهُمْ أَبْطَؤُوا أَخَّرَ
“Whenever he saw the people assembled (for the ‘Isha’ prayer) he would pray early and if the people delayed, he would delay the prayer.” 
Al-Hafiz Ibn Hajar said: There is also a narration from Muslim bin Ibrahim which shows that the Prophet PBUH waited until the companions and others have gathered to perform the prayer in congregation. This is prioritized over hastening to perform prayer (with fewer congregants). However, certainly one must consider whether by delaying it, there is no harm or burden for the congregants who are already present.