Is it obligatory for the mother to inform her aunt about the truth of the excess amount of money she received by mistake, or should she conceal it to avoid embarrassment, especially after she has spent part of it and allocated another part of it as a gift?

If it is highly probable that the money reached the mother by mistake from her sister (your aunt), then the mother must explain to her sister what happened. If the sister absolves her, there is no blame. Otherwise, she must return it to her, for the Prophet, peace and blessings be upon him, said: "On the hand is what it took until it returns it." The mother's desire to help her aunt's granddaughter does not waive her sister's right to the money, and it was not permissible for the mother to dispose of it before knowing the truth of the matter.