All police officers shall have the authority in criminal causes returnable before the Rutland District or Superior Court, to serve and return within the State said criminal process. For such duties the Police Department shall be allowed the fees provided by law to sheriffs for similar services, which fee shall belong to the City, to be paid into the City Treasury whenever such police officers shall be under pay from the City. The said police officers shall have the power of constables in all matters arising under the criminal and police laws of the State and ordinances and police regulations of the City.