Is the Fajr prayer offered immediately after the Adhan or after the Iqamah in the mosque?

If the muezzin gives the adhan after the time for prayer has begun, then there is no harm in praying immediately after the adhan. However, if he gives the adhan before the time for prayer has begun, then prayer is not permissible until after the time has entered. If the first adhan the questioner refers to is the one before true dawn (Fajr As-Sadiq), then it is not permissible, and she must make up those prayers. But if what is meant is the Fajr adhan that occurs after the dawn has broken, then the prayer is valid and she is not liable for anything.