Looking somewhat like a beret, that would be a Scot's bonnet, or tam-o'-shanter. COULD BE A GLENGARRY
Fez
could be a watch cap or a skullcap maybe even a slough hat
The Scottish brimless wool cap is called a "Tam o' Shanter," often simply referred to as a "Tam." It features a rounded, flat top and is typically made from wool, often patterned with tartan. The cap is named after a character in a poem by Robert Burns, and it is a traditional part of Scottish attire.
* rounded brimless cap fitting the crown of the head * the dome of the skull
Liberty cap is a type of hat. It is a soft brimless conical cap.
The hat is called a fez. It is normally found on the souks of Morocco and worn by men on special occasions.
The distinctive headgear of a shriner is a red fez, a kind of brimless conical cap.
The 'cap' (beanie) worn by some priests, and all bishops, cardinals and the pope is called a zucchetto.
A Fez
toque
A clergyman's cap is called a biretta.
In the book "Schooled," Sophie Donnelly used to call Cap, the main character, "Flower Boy."