Suppose we have a group $\Gamma$ acting on an abelian group $V$. Then it is well-known that the second cohomology group $H^2(\Gamma,V)$ corresponds to equivalence classes of central extensions of $\Gamma$, or equivalently, equivalences classes of short exact sequences of the form $$1 \longrightarrow V \longrightarrow E \longrightarrow \Gamma \longrightarrow 1$$ This interpretation can also be stated in terms of a lifting property: $H^2(\Gamma,V)$ comprises "obstacles" to the lifting of a homomorphism from $\Gamma$ to a quotient $W/V$ to a homomorphism from $\Gamma$ to $W$, where $W$ is another abelian group containing $V$.

Is there any such natural interpretation of the

bounded cohomologygroup $H_b^2(\Gamma,V)$ in terms of obstacles to lifting?

I know that when $V=\mathbb{R}$ and the action of $\Gamma$ is trivial, then the kernel of the comparison homomorphism $$c: H_b^2(\Gamma,\mathbb{R}) \to H^2(\Gamma,\mathbb{R})$$ comprises the space of non-trivial quasi-homomorphisms.

But what do the elements of the group $H_b^2(\Gamma,V)$ themselves represent for arbitrary $V$? I have been unable to find any satisfactory interpretation in existing literature so far.