$\mathbb{F}_p$ algebra with many $p$th roots of unity An old qual problem reads

Let $\mathbb{F}_p$ denote the finite field of $p$ elements. Consider
  the covariant functor $F$ from the category of commutative
  $\mathbb{F}_p$ algebras with multiplicative identity to abelian groups
  sending a ring $R$ to the $p$th roots of unity. 
  
  
*
  
*Give an example of a finite local ring $R$ such that $F(R)$ has $p^2$    elements.
  
*Let $\operatorname{Aut}(F)$ be the set of natural transformations of    $F$ into itself inducing a group automorphism of $F(A)$ for all 
  commutative rings $A$ with identity. Prove that $F$ is representable
  and use the Yoneda Lemma to compute the order of
  $\operatorname{Aut}(F)$.
  

For part a, what jumps to mind immediately is the group ring $\mathbb{F}_pP$ for $P$ an abelian $p$ group, which will be local. But the condition on the number of $p$th roots is giving me some trouble. I see that $A$ can't be reduced, otherwise there would be at most $p$ such roots. Any ideas?
Does someone have a suggestion for a reference where I can understand the basics of such algebras better? In particular rings which aren't semisimple.

$F$ is representable by the algebra $\mathbb{F}_p[x]/(x^p - 1)$. Since $x^p - 1 = (x - 1)^p$, this algebra is isomorphic to $\mathbb{F}_p[x]/(x - 1)^p$, which is in turn isomorphic to $A = \mathbb{F}_p[x]/x^p$. 
Now consider $R = \mathbb{F}_p[x]/x^3$. This is local. Homomorphisms $A \to R$ correspond to elements $r \in R$ such that $r^p = 0$. Elements of $R$ have the form $a + bx + cx^2$, and their $p^{th}$ powers have the form $a + bx^p + cx^{2p} = a$. Hence the elements such that $r^p = 0$ are precisely the elements such that $a = 0$, so there are $p^2$ such elements as desired.
Similarly, the elements $a \in A$ such that $a^p = 0$ are precisely the elements with zero constant term. Each such element corresponds to a morphism $A \to A$ (and vice versa). This morphism is an automorphism iff the linear term of the corresponding element is nonzero (exercise). This gives that $\text{Aut}(A)$ (and hence, by the Yoneda lemma, $\text{Aut}(F)$) has order $(p - 1) p^{p-2}$. 
