All perfect praise be to Allah, The Lord of The Worlds, and may His peace and blessings be upon our Prophet Muhammad and upon all of his family and companions.
Jurists of the four doctrines are of the view that a wife should observe her Iddah in the house in which the news of her husband's death came to you; whether it was owned by the husband or not (rent), for Allah, The Almighty Said (What means): "Expel them not from their houses nor let them go forth unless they commit open immorality." [At-Talaq/1].
Moreover, when the husband of Abi Saeed Al-Khodry's sister (May Allah be pleased with them) was killed, she asked for Prophet's (PBUH) permission to go back to her family because her husband didn't leave her a house nor an expenditure. But the Prophet (PBUH) refused and said (What means):"Stay in the house in which the news of your husband's death came to you, until your waiting period is over."' She said: "So I observed the waiting period there for four months and ten (days)." [Related by Abu Dawood No. 2300].
Further, the woman observing Iddah and lives in a rented apartment deserves to be given the fee of the rent out of her husband's inheritance until her waiting period is over in accordance to the Sharia ruling by which she is obliged to stay at her home during her Iddah. In addition, some scholars are of the view that a residence has to be provided for her while observing Iddah.
Therefore, the woman observing Iddah due to husband's death as mentioned in the above question, is obliged to stay at the marital home regardless of its location and to avoid travel Iddah period, because she isn't compelled to do so as her Mahrams live near her, since jurists are of the view that the travel of a woman during Iddah is prohibited. Yet, if she did so, then she is obliged to ask Allah for forgiveness and repent from such a sin as Allah The Exalted accepts the repentance of His slaves and pardon their sins. That’s why she is obliged to go back to the marital home because it is the only place in which her Iddah period can be observed .
In this regard, Al-Imam As-Shrbini (May Allah have mercy on his soul) stated: "The woman who didn't observe her Iddah period is obliged to go back to the marital home to observe her Iddah because it is the only place for making it up." [Mughni Al-Muhtaj/ vol.5 pp.108] And Allah Knows Best.