  We study the generic theory of algebraically closed fields of fixed positive
characteristic with a predicate for an additive subgroup, called
$\mathrm{ACFG}$. This theory was introduced recently as a new example of
$\mathrm{NSOP}_1$ non simple theory. In this paper we describe more features of
$\mathrm{ACFG}$, such as imaginaries. We also study various independence
relations in $\mathrm{ACFG}$, such as Kim-independence or forking independence,
and describe interactions between them.
