Transfer to an abelian group

Definition
Let $$G$$ be a finite group and $$H$$ and $$K$$ be subgroups such that $$K \triangleleft H$$ and $$H/K$$ is abelian and $$H$$ has finite index in $$G$$. Let $$T$$ be a left transversal of $$H$$ in $$G$$. Then define the following mapping $$V: G \to H/K$$

$$V(x) = \prod_{t \in T} h_x^T(t)K$$

here $$h_x^T(t)$$ is the unique element $$h \in H$$ such that $$xt = t'h$$ for some $$t' \in T$$.

We need to quotient out by $$K$$ so that the product on the right side is independent of the order of terms in the transversal.

Homomorphism
The transfer is a homomorphism of groups from $$G$$ to $$H/K$$.

Independence of choice of transversal
The transfer map does not depend on the choice of transversal $$T$$.