Pentactions and action representability in the category of reduced groups with action

A notion of pentaction any object in the category $\mathbf{rGr}^{\bullet}$ reduced groups with action is introduced. The operations are defined set $\mathsf{Pentact}(A)$ pentactions an $A$ $\mathbf{rGr}^{\bullet}$. It proved that if perfect zero weak stabilizer sense paper, then $\mathbf{rGr}^{\bullet}$, it has a derived on $A$, representable and represents all actions $A$.