What to call this extension of a linear transformation to the whole exterior algebra, defined by $F(A\wedge B)=F(A)\wedge B+A\wedge F(B)$ Given a linear transformation $f$ on a vector space $V$, we can extend it to the exterior algebra $\Lambda V$ by defining $F(A\wedge B)=F(A)\wedge F(B)$, or
$$F(1)=1$$
$$F(a)=f(a)$$
$$F(a\wedge b)=f(a)\wedge f(b)$$
$$F(a\wedge b\wedge c)=f(a)\wedge f(b)\wedge f(c)$$
$$\vdots$$
This is sometimes called the "outermorphism" of $f$; or its restriction to a particular grade, an "exterior power" of $f$.

But some situations (such as the rotational inertia tensor) suggest a different extension, defined by $F(A\wedge B)=F(A)\wedge B+A\wedge F(B)$, or
$$F(1)=0$$
$$F(a)=f(a)$$
$$F(a\wedge b)=f(a)\wedge b+a\wedge f(b)$$
$$F(a\wedge b\wedge c)=f(a)\wedge b\wedge c+a\wedge f(b)\wedge c+a\wedge b\wedge f(c)$$
$$\vdots$$
Does this extension have a standard name? If not, what do you think it should be called?
If nothing better appears, I'm considering "additive extension", as opposed to the outermorphism which is the "multiplicative extension".

Using the concepts from this answer of mine, the extension's restriction to grade $k$ can be written as
$$f\wedge\frac{(\wedge\text{id})^{k-1}}{(k-1)!},$$
the exterior product of $f$ with an exterior power of the identity.
 A: Your construction appears in various places but doesn't seem to have a standard name. In Sergei Winitzki's book "Linear Algebra via Exterior Products" it is called the "1-linear extension of $f$ to $\Lambda(V)$" (see page 131 of his book). One can generalize it and define the $k$-linear extension of $f$ to $\Lambda^m(V)$ which acts on $v_1 \wedge \dots \wedge v_m$ as a sum of ${m \choose k}$ terms which are obtained from $v_1 \wedge \dots \wedge v_m$ by choosing a subset $I \subseteq [m]$ of size $k$ and replacing each $v_i$ for $i \in I$ with $f(v_i)$. For example, the $m - 1$-extension of $f$ to $\Lambda^m(V)$ acts by
$$ v_1 \wedge \dots \wedge v_m \mapsto \\v_1 \wedge f(v_2) \wedge \dots \wedge f(v_m) + f(v_1) \wedge v_2 \wedge f(v_3) \wedge \dots \wedge f(v_m) + \dots + f(v_1) \wedge \dots \wedge f(v_{m-1}) \wedge v_m. $$
Such expressions arise naturally by taking derivatives of the full exterior power map (which you call outermorphism). They are also discussed in Chapter 6 of Werner Greub's "Multilinear Algebra" under the framework of "mixed exterior algebra" which is 
$$\Lambda(V^{*},V) := \Lambda(V^{*}) \otimes \Lambda(V) \cong \Lambda(V)^{*} \otimes \Lambda(V) \cong \operatorname{Hom}(\Lambda(V), \Lambda(V)). $$
In Greub's notation, the action of your map on $\Lambda^m(V)$ is denoted by
$$ f \square \underbrace{\operatorname{id} \square \cdots \square \operatorname{id}}_{m - 1 \textrm{ times}}. $$
