Let G=A⋊B, where A and B are abelian, and of coprime order. It seems, from my computations (and correct me if I'm wrong), that Z^{1}(C_{p},C_{q}) is trivial, for p and q different primes. Meaning that the automorphisms of G, if A=C_{q}, and B=C_{p}, that preserve A, and preserve the cosets G/A, are all trivial. How far can we extend this? Would it be true in general that Z^{1}(A,B) is trivial, with the above assumptions (that A and B are abelian and of coprime order)? If not, under what assumptions is it trivial? And when can we say about it if it's not trivial?

1$\begingroup$ No, I mean twisted as well. Otherwise it would just be direct product. $\endgroup$ – Terry Mahr Feb 18 '10 at 19:17

$\begingroup$ added tag "groupcohomology". $\endgroup$ – Daniel Moskovich Feb 25 '10 at 3:18
The automorphisms of this extension are basically the same as group cohomology $H^1(B; A)$, so I will focus on that first.
So we want to show that this must be zero given that B and A are finite abelian and of coprime order. This is the same as the twisted cohomology of the classifying space BB. Now we can look at the bundle $p:EB \to BB$. This is a covering space with fiber the discrete space B.
Now because the fiber is discrete we have a wrong way transfer map in twisted cohomology:
$$p_{!}: H^1(EB; A) \to H^1(BB; A)$$
where the first group is twisted cohomology in the pulledback local coefficient system. As with all transfers we have that
$$p_{!} \circ p^*: H^1(B; A) \to H^1(B; A)$$
is multiplication by the order of the fiber, i.e. $B$. Since the orders of A and B are coprime this is an isomorphism. But since $EG \simeq pt$ is contractible, this map factors through the zero group and hence $H^1(B; A) = 0$.
Any proof that the cohomology of a group is torsion for the order of the group (there are more concrete ones then the above) will yield the same result that $H^1(B; A) = 0$. There are many ways to prove this (as the comments point out), the above is just my favorite.
So what is the difference between the $H^1(B;A)$ and the isomorphisms of the extension G? Well as you pointed out the isomorphisms of G (which restrict to the identity on A and the quotient B) are the same as the bar resolution coycycles Z^1(B;A). So we have an exact sequence,
$$A= C^0(B;A) \to Z^1(B; A) \to H^1(B;A) \to 0$$
but as we saw, the term $H^1(B;A) = 0$. So we must compute the boundaries. You have one such potential homomorphism for each element of $A$, although different elements might give rise to the same automorphism of $G$. They are of the form:
$$b \mapsto a  b \cdot a$$
where $a \in A$ is fixed. If the action of B on A is trivial, then these vanish, but in general they can be nonzero. My favorite example is the quaterion group which we view as
$$\mathbb{Z}/4 \to Q_8 \to \mathbb{Z}/2$$
with the $\mathbb{Z}/4$ the group $( 1, i, 1, i )$. An element $x \in A = \mathbb{Z}/4$ induces the homomorphism
$$y \mapsto x  y \cdot x$$
which sends the nontrivial element of $\mathbb{Z}/2$ to $2x$. In particular it is nontrivial for a generator of A. This corresponds to the isomoprhism of $Q_8$ which sends $i$ to $i$ and $j$ to $j$.

$\begingroup$ Indeed, the vanishing of $H^1$ in this case follows from the standard study of the composition of restriction and correstriction maps; cf. HiltonStammbach, for example./ $\endgroup$ – Mariano SuárezÁlvarez Feb 18 '10 at 19:47

You might be considering a special case of the SchurZassenhaus theorem. If A is a normal Hallsubgroup of G, then A has a complement B, and all complements B are conjugate under the action of G. This is more properly H^1(B,A) rather than Z^1(B,A).
For Z^1(B,A) to be trivial, B^1(B,A) must be trivial, but B^1(B,A) is basically [B,A], which could be nonzero. For instance if G is nonabelian of order 6, then A is cyclic and normal of order 3, B is cyclic of order 2, and B acts as inversion on A. Then for B^1(B,A) should be isomorphic to A, that is have order 3. Basically, A has 3 complements in G, so B^1(B,A) should have three elements, but all are conjugate, so H^1(B,A)=0.
I suppose there are lots of definitions of C^1(B,A), so maybe you've chosen one where B^1(B,A)=0, but I think the standard choice when looking at semidirect products ("Crossed homomorphisms"), will not have B^1(B,A) trivial.