Group of prime power order admitting a bijective quasihomomorphism to an abelian group

Definition
A group of prime power order $$G$$ is termed a finite group admitting a bijective quasihomomorphism to an abelian group if there is an abelian group (in particular, an defining ingredient::abelian group of prim power order) $$H$$ and a bijective function $$f:G \to H$$ that is a quasihomomorphism of groups: whenever $$g_1, g_2 \in G$$ commute, we have:

$$f(g_1g_2) = f(g_1)f(g_2)$$