Which department is in charge of the social security audit department?

By the human resources and social security departments.

Baidu Encyclopedia says that the social security inspection office is set up by the human resources and social security departments, so it belongs to them.

Social security audit is a special activity carried out by the social security department to prevent some enterprises from underpaying and falsely reporting the social security base, protect the rights and interests of workers and effectively ensure the value-added security of funds.