Semidirect product of length two Witt ring and additive group

Definition
Suppose $$K$$ is a field of characteristic equal to a prime number $$p$$. The semidirect product of length two Witt ring and additive group is a three-dimensional algebraic group defined over $$K$$ as follows. It is the external semidirect product $$A \rtimes B$$ where:


 * $$A$$ is the additive group of the truncated ring of Witt vectors over $$K$$ to length two.
 * $$B$$ is the additive group of $$K$$.
 * The action is as follows. For an element $$(x_0,x_1) \in A$$ (note that elements of $$A$$ are represented by pairs of elements from $$K$$) and an element $$b \in B$$ (with $$b \in K$$), we define:

$$b \cdot (x_0,x_1) = (x_0,x_1 + bx_0)$$

where the multiplication and addition on the right in the second coordinate happens within the field $$K$$.

Particular cases

 * In the case that $$K$$ is the prime field $$\mathbb{F}_p$$, the additive group of the length two Witt ring is the cyclic group of prime-square order and the semidirect product becomes semidirect product of cyclic group of prime-square order and cyclic group of prime order.