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

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