Supercharacter theories for cyclic group:Z4

This article describes various supercharacter theories for cyclic group:Z4. We take the group a the integers mod 4, i.e., $$\mathbb{Z}/4\mathbb{Z}$$, and denote its elements $$0,1,2,3$$ with addition mod 4.

Summary
Each row represents a supercharacter theory. We describe the superconjugacy classes.

Automorphism group subgroup actions
The automorphism group of cyclic group:Z4 is cyclic group:Z2, with the non-identity element acting via the inverse map. The action of subgroups of this give supercharacter theories:

Galois group actions, or supercharacter theories based on character theories over subfields of the splitting field
The group cyclic group:Z4 has a unique minimal splitting field that is the cyclotomic extension $$\mathbb{Q}[i]$$. This is a quadratic extension over $$\mathbb{Q}$$. The Galois group of the extension is cyclic group:Z2.

Normal series
Below are the possible normal series and the associated supercharacter theories: