Is the salary that the girl received during her leave from work for treatment of her psychological illness permissible or impermissible, given that she remained at home and did not go to the syndicate she had been assigned to?

The employee's actions are governed by the contract concluded between him and his employer, as Allah Almighty says: "O you who have believed, fulfill [your] contracts," and the Prophet, peace and blessings be upon him, said: "Muslims are bound by their stipulations." So, if the decision-makers in the syndicate allow you to be absent from work due to illness, there is no harm in receiving your salary during your sick leave. And syndicate work must be based on truth and repelling injustice, and it is not permissible if it leads to defending falsehood.