Fix a finite group $G$. We call a pair $(K,(\sigma_g)_{g\in G})$

$G$-transformal field if $K$ is a field and the map

$$G\ni g\mapsto\sigma_g\in\aut(K)$$

is a homomorphism. In \cite{nacfa} we study the model companion, denoted by $G-\tcf$, of the theory of $G$-transformal fields.

It turns out that for an existentially closed $G$-transformal field the invariants of $G$ form a subfield, which is pseudo-algebraically closed in a rather strong sense. This leads to super-simplicity of the theory $G-\tcf$ and to few other nice properties.

Nevertheless, existentially closed $G$-transformal fields are not separably closed, and therefore $G-\tcf$ differs in an essential way from ACFA. Such difference fields belong to the class of fields with a group scheme action. The model theory of fields with operators coming from group schemes is being currently developed and $G-\tcf$ plays an important role in it.