I think perhaps that when one set is included within another it may be wise and more meaningful to include the curly brackets so that, for example, A is not mistaken for the alphabetic A, so B={{A}} rather than B={A}. Notation is only useful if it conveys precisely what is meant. So the double curly brackets may convey better the inclusion of a set. But, as far as I know (and I’m no set expert!), this is just an invention rather than a convention, so there may not be an official representation for what you want. A qualified, knowledgeable college or university tutor may know.
I always think it’s a good idea to explain notation (even if it’s an invention) to avoid ambiguity or misunderstanding, unless the notation is common knowledge. For example, because most input devices can’t typographically deliver “integral of f(x)dx with respect to x between the limits a and b”, I usually explain that I will be using “∫[a,b]f(x)dx” to represent it.
If A and B are sets, A=B means the sets are identical. If A={1 2 3} and B={4 5 6}, C={{A} {B}} could in this case be written C={A∪B} or C=A∪B, the latter probably being the standard option. So using set unions may be a better and more standard way of expressing the sort of thing you are looking for in some cases.
Sorry I can’t be more helpful.