Upon written request made by the managing authority of any referendum group, other than those referendum groups mentioned in section 68-621, and delivered to the Governor, the Governor, shall, within fifteen days after the receipt of such request, appoint the managing authority, or such other agency or individual, as he may designate, to conduct a referendum as provided by the provisions of sections 68-621 to 68-630 among the eligible employees included in such referendum group within four months of the date of such appointment. The cost of such a referendum shall be paid by the managing authority making the request.