  We lift to equivariant algebra three closely related classical algebraic
concepts: abelian group objects in augmented commutative algebras, derivations,
and K\"ahler differentials. We define Mackey functor objects in the category of
Tambara functors augmented to a fixed Tambara functor $\underline{R}$, and we
show that the usual square-zero extension gives an equivalence of categories
between these Mackey functor objects and ordinary modules over $\underline{R}$.
We then describe the natural generalization to Tambara functors of a
derivation, building on the intuition that a Tambara functor has products
twisted by arbitrary finite $G$-sets, and we connect this to square-zero
extensions in the expected way. Finally, we show that there is an appropriate
form of K\"ahler differentials which satisfy the classical relation that
derivations out of $\underline{R}$ are the same as maps out of the K\"ahler
differentials.
