6 year old indifference to punishement and make believe comfort

Hello mums, desperately need some guidance with the 6 year old son. The good points are he is gentle and sensitive and in every day life he's great boy, hardly any trouble besides usual as you would expect from the child. Great achiever in school, social and easy going. The struggle im having is when it comes to matters serious enough to warrant a punishment. This generally means him loosing things he loves i.e favourite toy, TV time or anything of sort. He just doesn't seem to care or show that he does and takes it in as long as he doesn't have to admit he was wrong? I'd like to think im reasonable person and giving him every opportunity to get his point across and explain the reasons as to why he has done what he has or understand the process of thought while he has done it. As an example, there was an incident at school. There were 3 kids involved, but for some reason he doesn't admit that there were more than two of them including himself. The head teacher we spoke with had said that actually the parents of the child who he is not acknowledging  have come to school to discuss the interaction between those 3 kids ( including mine). And this is what I have noticed, he doesn't know how to lie, but when he doesn't like something badly enough or stresses him out, then he creates the scenario which is much more comfortable for him and makes believe this is real. However nicely you can dress this up this is still a lie. And for this he has lost some of his privileges. But yet again, he would not admit and rather wait out for the punishment to pass but not tell the truth. Please help, as really not sure how to get through to him and help him. Help him to understand, that running away from something or telling something that's I know it isn't true not the right way going about it. Had heart to heart with him, to which he seems to understand and acknowledge, have punished, and yet still back to square one.  Thank you