Who administers direct democracy mechanisms at the national level?

Cuba's Constitution of 1976 (last amended in 2002)

ARTICLE 74

The National Assembly of People’s Power elects, from among its deputies, the Council of State, which consists of one President, one First Vice President, five Vice
Presidents, one Secretary and 23 other members. The President of the Council of State is, at the same time, the Head of State and Head of Government.
The Council of State is accountable for its action to the National Assembly of People’s Power to which it must render accounts of all its activities.

ARTICLE 90

The Council of State is invested with the power to:

[…] e. make all the necessary arrangements for the holding of referenda called for by the National Assembly of People’s Power;

Council of State is responsible for administering the referendums. It consists of one President, one First Vice President, five Vice Presidents, one Secretary and 23 other members. The President of the Council of State is, at the same time, the Head of State and Head of Government.