The functions of the State Public Service Commission of Uttar Pradesh can be extended by

Under Article 321 of the Constitution of India, the power to extend the functions of a Public Service Commission lies with the respective legislature. Specifically, it states that an Act made by the State Legislature may provide for the exercise of additional functions by the State Public Service Commission (SPSC) regarding the services of the State. This extension can also cover the services of any local authority, other body corporate constituted by law, or any public institution within the state's jurisdiction.

Therefore, for the State Public Service Commission of Uttar Pradesh (UPPSC), the Uttar Pradesh State Legislature is the competent authority to extend its remit. Options A, B, and C are incorrect because the President and Union-level authorities only have the power to extend the functions of the Union Public Service Commission (UPSC) via Parliamentary acts, not the SPSC.