@MartinSleziak: This question allows me to test the thinking of generalizations as duplicates. The result surely follows from the general one I linked to. It did spawn a solution special to this case in that the only case we need to exclude is that of all the elements being of order three. But, all the answers come around to the same argument at the end - the kind of repetition I wanted to avoid having.

@JyrkiLahtonen I have cast the first vote. But my suggestion would be to wait and see how the reviewers in the review queue will react.

There is always a possibility to go over all through all groups of order 12. But that seems as an overkill. (And I do not think that it is an easy thing to do.) groupprops.subwiki.org/wiki/… math.stackexchange.com/questions/1583743/…

But also this answer (hint) seems to be specific for order 12: math.stackexchange.com/questions/1840159/… "Consider the Sylow $2$-subgroups of $G,$ which have order $4$."

Let's wait and see how this will go in the review queue or whether somebody will protest closing in comments or on meta or in chat.