  Derived actions in the category of groups with action on itself
$\mathbf{Gr}^{\bullet}$ are defined and described. This category plays a
crucial role in the solution of Loday's two problems stated in the literature.
A full subcategory of reduced groups with action $\mathbf{rGr}^{\bullet}$ of
$\mathbf{Gr}^{\bullet}$ is introduced, which is not a category of interest but
has some properties, which can be applied in the investigation of action
representability in this category; these properties are similar to those, which
were used in the construction of universal strict general actors in the
category of interest. Semi-direct product constructions are given in
$\mathbf{Gr}^{\bullet}$ and $\mathbf{rGr}^{\bullet}$ and it is proved that an
action is a derived action in $\mathbf{Gr}^{\bullet}$ (resp.
$\mathbf{rGr}^{\bullet}$) if and only if the corresponding semi-direct product
is and object of $\mathbf{Gr}^{\bullet}$ (resp. $\mathbf{rGr}^{\bullet}$). The
results obtained in this paper will be applied in the forthcoming paper on the
representability of actions in the category $\mathbf{rGr}^{\bullet}$.

Category Theory;Group Theory