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

Derived actions in the category of groups with action on itself Gr^ are defined and described. This category plays a crucial role in the solution of two problems of Loday stated in the literature. A full subcategory of reduced groups with action rGr^ of Gr^ 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 Gr^ and rGr^ and it is proved that an action is a derived action in Gr^ (resp. rGr^) if and only if the corresponding semi-direct product is an object of Gr^ (resp. rGr^). The results obtained in this paper will be applied in the forthcoming paper on the representability of actions in the category rGr^.