ANSWER:

What she is doing is baseless.

You can pray as soon as the time for isha enters, regardless of when the athan is given or when the iqamah or congregation begins in the masjid. 

When it is time to pray isha my quran teacher waits till we are done with the lesson. Which is 30 minutes or more after the iqama of isha was given. Then he tells us to pray. Is it permissble to do this?

Prayer |