The correct answer is option 'A', the Planning Commission.

Explanation:
The Planning Commission was a non-constitutional and non-statutory body that undermined the role of the Finance Commission in Centre-state fiscal relations. The Planning Commission was established in 1950 and functioned as the main policy-making body in India's centralized planning system.

Undermining the Role of the Finance Commission:
1. Allocation of Resources: The Planning Commission played a significant role in the allocation of financial resources between the Centre and the states. It determined the share of resources to be allocated to each state based on its own criteria, thereby undermining the role of the Finance Commission.

2. Discretionary Grants: The Planning Commission had the power to provide discretionary grants to states for specific projects or schemes. These grants were not determined by the Finance Commission and were often used by the central government to influence and control state governments.

3. Centralized Planning: The Planning Commission formulated five-year plans that set targets for economic growth and development. The allocation of resources under these plans was done by the Planning Commission, bypassing the Finance Commission's role in determining the distribution of financial resources between the Centre and the states.

4. Vertical Imbalance: The Planning Commission's approach often resulted in vertical imbalance in fiscal relations between the Centre and the states. This means that the Centre retained a larger share of financial resources, limiting the fiscal autonomy of the states.

5. Lack of Transparency: The Planning Commission's decision-making process lacked transparency and was not based on objective criteria. This undermined the principles of cooperative federalism and the role of the Finance Commission in ensuring a fair and equitable distribution of resources.

Replacement by NITI Aayog:
To address the issues associated with the Planning Commission, it was replaced by the NITI Aayog (National Institution for Transforming India) in 2015. The NITI Aayog is a policy think tank that provides strategic and technical advice to the central and state governments. Unlike the Planning Commission, the NITI Aayog does not have the power to allocate financial resources or provide discretionary grants, ensuring a more balanced and transparent approach to Centre-state fiscal relations.