George III was a constitutional monarch: an executive constrained by laws and institutions. For the most part, the so-called constitutional monarchs of today are in fact symbolic. Except when there’s a hole in the constitution; then the ghost of past power re-emerges and Kings, Queens and Governors-General of Canada or Australia actually get to decide something, like a president of Italy or Germany.
The live case is when a government loses its majority in the legislature. New election, or try to form a new government under a different prime minister?
In the United Kingdom, the only authoritative guidance is, believe it or not, offered by an anonymous letter to The London Times in 1951 by the King’s Private Secretary, Sir Alan Lascelles. Lascelles reckoned that the monarch should try the second option. But it hasn’t happened yet.
PS: the Governor-General of Canada, Michaëlle Jean, is a black woman of (bourgeois) Haitian origin. So Canada beat the United States to it on two counts.