How to randomly select a row from each group in MySQL

Suppose I have a table like this:

item   | group
-------+--------
item a | group z
item b | group y
item c | group y
item d | group x
item e | group z
item f | group x

And now I want to randomly select a single item to represent each group, so that after random reordering and grouping, the result looks like this:

item   | group
-------+--------
item b | group y
item d | group x
item e | group z

Or so:

item   | group
-------+--------
item b | group y
item a | group z
item d | group x

Or so:

item   | group
-------+--------
item f | group x
item e | group z
item c | group y

All scrambled and grouped. How can I do this?

PS1: Note that: - time the group x is associated with the item d and formerly to item f; - time the group y is associated to item b and formerly to item c; - time group z is associated to item a and to item e;

PS2: If it is possible to follow the line of reasoning for this post here, it is better: Select the first row of each group in MySQL otherwise, no problem at all!

PS3: I've seen similar explanations using the HOVER and PARTITION command. But these commands do not exist in MySQL. So I really need help.