Is it permissible for me to take money from my sister, who works in Saudi Arabia, to invest it as if it were my own, and then return the profit and principal to her without her husband's knowledge?

There is no harm for your sister to lend you money as a loan for you to invest and benefit from its returns, provided that the capital remains hers. However, it is not permissible for her to share in the profit while keeping the capital, as that constitutes usury (riba). Her husband's knowledge of her giving you the money is not a condition, as her money is her own property, and he has no right to it except with her consent, for Allah the Almighty says: (And give women their dowries with a good heart; but if they, of their own good pleasure, remit any part of it to you, take it and enjoy it with good cheer and satisfaction). However, it is advisable to inform the husband due to his experience in trade, and for the sake of good companionship. The wife owns her money and has the right to dispose of it without her husband's objection if she is of sound judgment.