Automorphism-invariant supercharacter theory

Definition
A supercharacter theory is termed automorphism-invariant if it satisfies the following equivalent conditions:


 * 1) Its blocks of conjugacy classes are permuted among each other by any automorphism of the group, i.e., any automorphism of the group sends a block to a block.
 * 2) Its blocks of irreducible characters are permuuted among each other by any automorphism of the group, i.e., any automorphism of the group sends a block to a block.