I concede that the Friesland lists of ships with dimensions and gun lists would lead you to believe that the ship Graaf Hendrick, commanded by Jan Reijndersz Wagenaer in late 1652 and all of 1653 was a ship hired by the Admiralty of Friesland. However, there are some other, independent lists, that label that ship as a ship hired by the Groningen Directors.
In a list from 18 May 1652, Joost Bulter was listed as commanding a Groningen Directors' ship. My working theory, which seems more plausible now, is that Joost Bulter commanded the Graaf Hendrick up until July 1652 or so, when Jan Reijndersz Wagenaer was appointed in his place. Joost Bulter was nominated as the captain of a new ship purchased for the Admiralty of Friesland which was eventually named the Stad Groningen en Ommelanden.
I now have seen three unrelated lists that show that the Graaf Hendrick was employed by the Groningen Directors in November and December 1652. It remains to be seen if in fact Joost Bulter commanded the ship or if he was just listed as potentially commanding a ship. As for the Graaf Hendrick, the lists of dimensions and guns usually say that it was a ship of Groningen and leaves ambiguous whether it was employed by the Groningen Directors or by the Admiralty of Friesland. I expanded upon what I originally wrote after receiving some comments.