If you are on good enough terms with the boy's mother, you could ask her if the child has understood her correctly when she has said no to these activities.
Mind you, a boy of six should not be taking karate classes - he is far too young and there are far too many irresponsible karate teachers, who do not take trouble to explain that all martial arts should be used for self-defence rather than to hurt others.
As others have said, she may well feel she cannot afford these activities, or she may know that his school-work is so tiring, or so poor that time is better spent either doing homework, or playing after school and getting to bed at a reasonable hour.
You could as already suggested offer to help out with payment, but ask her, don't just go on what a six year old says, his mother says.
But be tactful, otherwise you risk being asked to mind your own business.