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.