Is it permissible for the work supervisor to take a portion of the work to execute it himself as a subcontractor for his own account, knowing that he is responsible for receiving it and ensuring it meets specifications? And is this considered unlawful or lawful?

What the overseer and the contractor did constitutes bribery, for bribery is what is given to nullify a right or to uphold falsehood. The contractor's intention behind the payment was to have his deficiencies overlooked and to be leniently treated in the reports. The supervisor is blameworthy, and it is impermissible for him to work with the contractor without the permission of his employer, because he is their agent in receiving the work, and thus it is forbidden for him to receive it from himself, except with the employer's permission.