Greenland has been a part of the Kingdom of Denmark since 1814. Over the years it has been granted more and more autonomy, but it is still under Danish rule. In fact, the Danish monarchy retains more control over Greenland than the British monarchy retains over Australia.
I'm not sure how you would define the Danish monarchy, but I am pretty sure that King Frederik has no control over Greenland whatsoever. Much the same way that Greenland has been granted more and more autonomy over the years, Danish kings and queens have lost control of The Danish Realm. Control of the Danish Realm is in the hands of Folketinget, i.e. the Danish parliament, and the parliaments of Greenland and the Faroe Islands.
The point of royal families in constitutional monarchies is to represent their respective countries
symbolically and
only symbolically. They are
symbols of nationalism, which only work as such if and because they stay out of politics. As such, they serve to
unite people in a country divided by economic, social and political interests.
It becomes most apparent on New Year's Eve when the speeches of two people are broadcast to the nation: the prime minister's and the king's (currently) or queen's (for the past more-than 50 years). Half of the people listening to those speeches (some years a little more, some years a little less) can't stand whoever is prime minister at the time because of his/her politics. But more or less everybody (I'm not one of them) likes the monarch's speech because the monarch stays out of politics,
has to stay out of politics, and makes no decisions.
They have something similar in some democracies that aren't monarchies. In Germany, for instance, they have a Bundeskanzler (chancellor) and a Bundespräsident. The former is the head of government, i.e. the prime minister. The latter does what monarchs do in constitutional monarchies: represent the nation but without much actual power:
What is the difference between a chancellor, e.g. German chancellor and a president? (Quora - Daniel Plomp)
In the USA, it is apparent that there is no such symbolic figure uniting the nation. The rift between the class that makes political decisions, i.e. the oligarchs, and the people who are governed by that class has become too big to be bridged by the presidency even though he does represent both sides - in a way: He issues policies that serve the oligarchs, but he presents himself as a 'man of the people,' an enemy of the elites, to his fans, but since the people whose 'man' he is is defined by race and not all white people are old-fashioned racists anymore, it only serves to exacerbate the already existing rift.
King Frederik can go to Greenland and represent The Danish Realm and be
accepted by Greenlanders as the representative of The Danish Realm because he doesn't get to make any policy decisions, i.e. because he has no actual control of anything.