Possibly. Many times, local taxing authorities reduce the income only by any amounts in Box 12 Code C of your Form W-2. But, can add to the income from other items in Box 12. Likewise, there could be a special treatment for income earned from another state.
To determine if you are allowed to reduce the local income reported to only 1/5 when working out of state, you need the local income tax instructions.
To start, check the local form for any instructions. Often times on tax returns for town, directions are more integrated with the form. Otherwise, to find local income tax instructions, I recommend going to your local government office or website.
Lastly, it may be useful to discuss the withholding with your employer to either see about getting a corrected W-2 or adjusting any reporting necessary for following years.
