I am aware that Islam encourages looking after the orphans. It also prohibits giving your own lineage in place of their original lineage (through a change in surname).
But what is the ruling related to giving them inheritance after our death?
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Those people whom can inherit are described and listed in a number of Verses of the Qur'an. Among them you'll find the (bodily) children, (biological) parents. grandparents and wife/husband, (bodily) sisters and brothers!
But adopted children don't count because of the Verses (33:4-5) which made Adoption (by naming a child with your name) illegal or void! But sponsoring them is allowed!
So inheritance in therm of irth إرث or ميراث wouldn't be permissible for adopted children!
I only could find fatwas in Arabic on the matter 1 and 2 and would like to add an info from the last one, if the inheritance has been divided and the Adopted child had got some of it and there was no last will in his/her favour he/she must give back what he/she got and that would be divided between the heirs!
And Allah knows best!