Is it Islamically permissible to register one's son and wife as employees in an institution without them actually joining the work, for the purpose of obtaining the wages of two additional employees?

This offer from the institution's manager is not permissible শরعًا (religiously) and constitutes consuming wealth unjustly and betraying the institution, unless the manager is the actual owner of the institution, in which case it would be considered charity (صدقة) or a gift (هبة). If the manager is taking a bribe and demanding payment for what he offered, then the sin is compounded by adding the sin of bribery to the sin of betrayal, because he is an employee entrusted with the institution's funds, and this action of his is unlawful (haram) and a betrayal of trust. Allah Almighty says: "O you who have believed, do not betray Allah and the Messenger or betray your trusts while you know [the consequence]."