In this note we study check character systems (with one control symbol) over groups (over abelian groups) and the check formula a1 ·δa2 ·δ2a3 · · · · ·δnan 1 = e, where e is the identity of a group, δ is an automorphism (a permutation) of a group. For a group we consider strongly regular automorphisms (anti-automorphisms), their connection with good automorphisms and establish necessary and sufficient onditions in order that a system to be able to detect all single errors, transpositions, jump transpositions, twin errors and jump twin errors simultaneously.