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 Z1(Cp,Cq) is trivial, for p and q different primes. Meaning that the automorphisms of G, if A=Cq, and B=Cp, 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 Z1(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?