Is it permissible to lend money to a female relative without her husband's knowledge, given that there is no risk of temptation due to the distance between them?

There is no objection to lending money to the paternal aunt's daughter without her husband's knowledge, as long as the transaction is within the bounds of Sharia and there is no fear of fitna (mischief or discord). Rather, the doer will be rewarded for this, as the Prophet, peace and blessings be upon him, said: "Whoever relieves a believer of a worldly grief, Allah will relieve him of a grief on the Day of Resurrection. Whoever facilitates for a person in difficulty, Allah will facilitate for him in this world and the Hereafter. Whoever conceals (the faults of) a Muslim, Allah will conceal his faults in this world and the Hereafter. Allah is in the aid of His servant as long as His servant is in the aid of his brother."