Is it permissible for me to help my mother with the expenses of Hajj using money my husband's father gave me for investment, with the intention of saving the profit for my children, but without the permission of my husband or his father?

The money given to you by your father-in-law, if it was a gift and was taken possession of, then it is yours, and you may dispose of it as you wish. However, if it was for the purpose of Mudarabah (a partnership in profit), then the percentage of profit between you must be determined, and it is not permissible to dispose of the profit before the division of the capital, except with the permission of the owner of the capital.