They need to go to the Edit Profile page and select the group badge they want displayed:
The animation below shows a slow click (which failed to navigate) followed by a fast click (which successfully navigated). It was recorded on the Qt Forum, but the same behaviour occurs on https://community.nodebb.org/ (Tested today, on Firefox 86.0 and Chrome 88.0.4324.190 on Windows 10)
I believe this occurs because a mouse-down event causes the pop-up to disappear. However, if the clickable link disappears before the user releases the mouse button, then the click event does not trigger.
You are correct, when the input element loses focus it hides the dropdown hence it might miss the click on the result if you click hold a while and release.
This should be fixed on master and on this forum as well.
@jksh What a bug, thanks for reporting!