Restricted regular wreath product of group of prime order and quasicyclic group

Definition
Let $$p$$ be a prime number. The restricted wreath product of group of prime order and quasicyclic group is defined as the restricted external wreath product of the cyclic group of prime order (i.e., the cyclic group of order $$p$$) by the $$p$$-quasicyclic group for the regular group action.

Note that the term restricted indicates that we use a restricted direct product rather than an unrestricted direct product in the base.

Equivalently, if $$H$$ denotes the $$p$$-quasicyclic group, then this group is the external semidirect product of the additive group of the group ring $$\mathbb{F}_p[H]$$ by the action of $$H$$ by left multiplication.