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We study the RKKY interaction mediated by the helical edge states of aquantum spin Hall insulator in the presence of the Rashba spin-orbital couplinginduced by an external electric field and the electron-electron interaction. Weshow that in the presence of the Rashba coupling, the RKKY interaction inducedby the helical edge states contains not only the Heisenberg-like and theDzyaloshinskii-Moria terms but also the nematic-type term that is not presentoriginally, with the range functions depending on the strength of the Rashbacoupling. We also show that the electron-electron interaction changes thestrength of the RKKY interaction by modifying the power of the $1/|x|$dependence of the range functions. In particular, by varying the strength ofthe interaction or the Rashba coupling, there is an (impurity) quantum phasetransition involving the sign change of the RKKY interaction at the value ofthe Luttinger liquid parameter $K=1/2$. Since both the strength of the Rashbacoupling and the chemical potential of the helical edge states are electricallycontrollable by external gate voltages, our results not only shed light on thenature of magnetic impurity correlations in the edge of a two-dimensionaltopological insulator, but also pave a way to manipulate the qubits in quantumcomputing.

Electrical control and interaction effects of the RKKY interaction in helical liquids