From the hadith we learn that the person who gives the Adhan should also give the Iqamah when Salat is about to start. It is his first right to give the Iqamah.

A hadith in this regard is:

Ziyad^(ra) bin Harith as-Suda’i narrated:

عَنْ زِيَادِ بْنِ الْحَارِثِ الصُّدَائِيِّ، قَالَ كُنْتُ مَعَ رَسُولِ اللَّهِ صلى الله عليه وسلم فِي سَفَرٍ فَأَمَرَنِي فَأَذَّنْتُ فَأَرَادَ بِلاَلٌ أَنْ يُقِيمَ فَقَالَ رَسُولُ اللَّهِ صلى الله عليه وسلم “‏إِنَّ أَخَا صُدَاءٍ قَدْ أَذَّنَ وَمَنْ أَذَّنَ فَهُوَ يُقِيمُ‏”‏

“I was with the Messenger of Allah on a journey, and he commanded me to call the Adhan. Bilal wanted to call the Iqamah, but the Messenger of Allah said: ‘The brother of Suda called the Adhan, and the one who calls the Adhan is the one who calls the Iqamah.’”

From this it is clear the one who calls the Adhan has the right to call the Iqamah too.

However, if for some reason the person who called the Adhan is not present, the Imam can appoint someone else to call it.