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Yes, the entity is considered a foreign entity and must register using form CT-245. Here is a short excerpt from the instructions for the form which can be accessed at www.tax.ny.gov:
Who must file Form CT-245
Any business incorporated outside New York State that is
authorized to do business in New York State and wishes to
disclaim tax liability.
qualified subchapter S subsidiary (QSSS) incorporated
outside New York State and authorized to do business in
New York State that is included in the parent corporation's
Any business incorporated outside New York State having an
employee (including any officer) in this state and disclaiming
tax liability. (Employees are considered to be working in this
state if they come into New York State on corporate business.)
I think you didn't answer what was asked. I spoke of a 501 c 6 entity which has been exempted from franchise tax but could be subject to not for profit reproting. Your answer is incorrect