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

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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 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)