All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
As regards repeating after the caller of Athaan, then this is prescribed for anyone who hears the Athaan, but if someone does not hear it, then it is not conceivable that one is then also required to do so. However, some scholars are of the view that when a woman wants to pray, it is permissible for her to call the Athaan and the Iqaamah; Ibn Qudaamah said: “Ash-Shaafi’i said: If women call the Athaan and the Iqaamah, then this is acceptable. Also, ‘Aa’ishah used to call the Athaan and the Iqaamah. This is also the view of Is-haaq. It was also reported that Umm Waraqah narrated that the Prophet permitted for her that the Athaan and the Iqaamah be called for her, and then she would lead the women of her household in prayer.” [End of Ibn Qudaamah's quote]
As regards repeating the Iqaamah, then according to the view of the majority of scholars it is prescribed to say, 'Aqaamaha Allaah wa adaamaha' when the Muaththin says ‘Qadi Qaamati As-Salah'. Although the Hadeeth is considered Dha'eef (weak), it is concerning acts of virtue and the weakness in such Ahaadeeth is tolerated in contrast to other Ahaadeeth. For more benefit, please refer to Fataawa 86128 and 88382.
However, some scholars are of the view that it is not prescribed to repeat the Iqaamah at all; this is the view preferred by Shaykh Ibn ‘Uthaymeen and the lay person should follow the view of the scholars he or she trusts most.
Allaah Knows best.