Related to Henry Wilton's comments: the following might not quite be what you're looking for, but seems interesting given that quasi-morphisms have been mentioned. I'm doing this from memory so if there's a gap, someone please let me know!
Let $E$ be a Hilbert space, $B(E)$ the algebra of all bounded linear operators on $E$. (Even the case $E={\mathbb R}^n$ is of interest.) Fix a small $\epsilon>0$. Then there exists $\delta>0$ with the following property:
Let $G$ be an abelian group, and let
$f:G \to B(E)$ be a bounded function (i.e. $\sup_{x\in G} \| f(x) \| < \infty$) which satisfies $$ \sup_{x,y}\| f(x)f(y) - f(xy) \| \leq \delta. $$ Then there is some representation $\rho: G \to B(E)$ such that $\sup_x \| f(x)- \rho(x)\| \leq \epsilon$.
So, less formally, bounded "almost representations" of abelian groups are "near to" genuine representations.
I imagine this could be proved by an averaging argument: the way I learned of this result is as a special case of a more general one, in which the word "abelian" is replaced by the word "amenable", and the word "Hilbert" is replaced by "nice reflexive Banach". That in turn is a special case of a general result on almost multiplicative maps between Banach algebras satisfying certain conditions (due to B. E. Johnson).
Anyway, sorry this has wandered off track. The point was to say that there are contexts where things which are close to being group homomorphisms $H\to K$, might under a small perturbation be genuine homomorphisms when restricted to a specified abelian subgroup of $H$. However, in general this can't be done so as to work simultaneously for all abelian subgroups of $H$.