Under fair use, I will quote Exercise 22, Section 5.5 from Dummit and Foote's Abstract Algebra. Take note that the topic of the section is semidirect products.
Let $F$ be a field let $n$ be a positive integer and let $G$ be the group of upper triangular matrices in $GL_n(F)$ (c.f. Exercise 16, Section 5.5).
(a) Prove that $G$ is the semidirect product $U \rtimes D$ where $U$ is the set of upper triangular matrices with $1$'s down the diagonal (cf. Exercise 17, Section 2.1) and $D$ is the set of diagonal matrices in $GL_n(F)$.
(b) Let $n=2$. Recall that $U \cong F$ and $D \cong F^\times \times F^ \times$ (cf. Exercise 11 in Section 3.1). Describe the homomorphism from $D$ into $Aut(U)$ explicitly in terms of these isomorphisms (i.e., show how each element of $F^\times \times F^ \times$ acts as an automorphism on $F$.
I have already solved part (a). One shows that $U \unlhd G$ and $D \leq G$ and $U \cap D=\{I\}$ and $UD=G$. This implies that $G \cong U \rtimes _\phi D$, where $\phi:D \to Aut(U)$ by conjugation.
For part (b), however, I am confused on what exactly is being asked for. (I understand why $U \cong F$ and $D \cong F^\times \times F^ \times$). Is it merely asking for an explicit homomorphism $h: F^\times \times F ^ \times \to Aut(F)$ which satisfies $G \cong F \rtimes _h (F^ \times \times F ^\times)$? Or is it asking instead for a homomorphism from $D \to Aut(G)$? In either case, do I need to show that the homomorphism is unique?
Take note that my question is more about "logistics" (trying to decipher instructions) than actual mathematics. However, any help with part (b) would be welcome and appreciated.