Is it permissible for the wife to give charity from the household allowance without her husband's knowledge, given his objection to it and his claim that he gives charity on behalf of both of them, but the wife does not believe him?

A wife is entrusted with her husband's wealth, so it is not permissible for her to dispose of any part of it without his permission, due to the Prophet's (peace and blessings be upon him) saying: "A woman is a guardian in her husband's house, and she is responsible for her charge," and his saying: "A woman should not spend anything from her house except with her husband's permission."

So, if the husband forbids her from giving charity from his money, it is not permissible for her to do so. Therefore, it is not permissible for you to give charity from your husband's money as long as he has forbidden you from doing so.