Actions and semi-direct products in categories of groups with action

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