Is it permissible to secretly give the amount that the sister willed to be given for the sake of Allah after her death to her indebted son, without the knowledge of his siblings?

The executor is not permitted to give the bequeathed charity to the heirs of the testator, regardless of whether they are rich or poor. This is because the executor acts on behalf of the deceased, and should not disburse funds to those to whom the one who appointed him would not disburse.

If the heirs do not believe the executor's claim that there is a will, the executor must provide lawful evidence to prove the will, or hand over the money to the heirs for them to divide according to Islamic inheritance law.

A will is not established for the executor except with two witnesses, whereas it is established for the legatee with one witness and an oath, or with one witness and two women.