What is the ruling on giving some money or food to one's married sister and her children secretly from the stepmother? Is this considered a betrayal of the father? Will the doer be rewarded for it? And is it equivalent to giving charity?

It is permissible to give charity to your sister from your own private wealth, for it is both charity and the upholding of kinship ties. However, it is not permissible to give charity from your father's wealth except with his permission. If he gives permission, his wife's disapproval does not matter. It is advisable to encourage your father and brothers to give charity to your sister due to the reward in it. If the father gives permission, he will have his reward, and you will have the reward for guiding to good.