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My pleasure, Mel. My first thought was Dutch, but after a bit of Googling I decided to try Swedish as their insignia often incorporates a crown motif. This is stated as a WW2 Swedish Air Force cap badge:
Note the similarity between the wings on this badge and those on your friend's coat.
Here's a more modern Swedish Air Force Kapten's shoulder board:
Yes, it's a badge of our Air Force (nowadays we refer to it as "the Air Force", previously it was called the Royal Swedish Air Force).
Once upon a time, during the 1950s, the Swedish Air Force was regarded as the 4th biggest Air Force in the world. Concerning that we at that time only had about 7 million inhabitants, it was a rather high "planes per capita" ratio!