Let $\langle x\rangle$ be a cyclic group, and $N$ any group. It is easy to tell when a map $x\mapsto n $ can be extended to a homomorphism: if $o(x)$ is infinite then always; if $o(x)$ is finite then it (map) can be extended to homomorphism only if $o(n)$ divides $o(x)$.
Let $H,N$ be groups with an action of $H$ on $N$ (i.e. homomorphism $H\rightarrow Aut(N)$). A map $f$ from a group $H$ to a group $N$ is called crossed homomorphism if $$f(hk)=f(h)^kf(k)$$ for all $h,k\in H$ where $f(h)^k$ denotes action of $k\in H$ on $f(h)\in N$.
The question is the following natural one:
given a cyclic group $\langle x\rangle$ with action on $N$, when the map $x\mapsto n$ can be extended to a crossed homomorphism?