Which among the following has the power to extend the functions of the State  Public Service Commission? 

As per Constitution of India, Art. 321 . Power to extend functions of Public Service Commissions. An Act made by Parliament or, as the case may be, the Legislature of a State may provide for the exercise of additional functions by the Union Public Service Commission or the State Public Service Commission as respects the services of the Union or the State and also as respects the services of any local authority or other body corporate constituted by law or of any public institution. Therefore, both Parliament and State Legislatures have power to extend functions of Public Services Commissions. In the above question only State Legislature is the apt option.