Here is some information about your cruet. You have the crowned A charge mark in use in Paris from 1768-1774, and the discharge mark for large objects - a woman’s head facing right - for the same period. The date letter H indicates that the object was made in 1771. The crown on the H has disappeared under the bolt attaching the holders to the plate.
As for the maker, initials JCG with a sun as symbol, I believe this is the mark of Jean-Charles Gordière. Gordière had a long career, and moved many times. He was apprenticed in 1735 at the age of 13 to Jean de Harchies. In 1747 and 1748 he registered his mark. In 1748 he was recorded as living rue St. Eloy, then he was listed on quai Pelletier in 1750 and 1758. By December 1759 he was at the Galeries du Louvre; in 1765 on the pont au Change; 1766 rue de la Tabeletterie; 1768 cul-de-sac St Martial; 1769 rue du Marché Palu; 1774 cour Lamoignon. In December 1777 he turned in his poinçon. Thus his working dates are 1747-1777, and your object probably was made at his rue du Marché Palu address. His many moves and low ranking on the tax list indicate to me that he worked mostly as a jobber for other silversmiths, but this is only conjecture.
See Nocq, v. 2, p. 266, the addenda for v. 2, p. 20, and the general index, p. 48.
As a bonus, you have a personalization, which appears to be of the period: a marriage crest under the 9-point crown of a count. Further research might tell you the name of the owners of your cruet.
Hope this helps.