On check character systems over groups

In this note we study check character systems (with one control symbol) over groups (over abelian groups) and the check formula $a_1\cdot\delta a_2\cdot\delta^2 a_3\cdots\delta^n a_{n+1}=e$, where $e$ is the identity of a group, $\delta$ 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 conditions in order that a system to be able to detect all single errors, transpositions, jump transpositions, twin errors and jump twin errors simultaneously.