Is it permissible for a woman to give part of her monthly allowance to her mother so that the mother can visit her sisters, give them gifts, and invite them, all without the husband's knowledge?

It is permissible to give charity to the mother, and this is a good deed due to the significant benefit it entails and its role in reconciling differences. There is no obligation to inform the husband about the specific use of the money if doing so would lead to harm. Dutifulness to parents is obligatory even if they treat one poorly, for their rights are great, second only to the right of Allah.