I'm using AEM6.4.4 and have created a user group and added couple of users in it. The issue is, when I double click on the user group, I'm able to see the user A that I have just added to the group but when I search for the user A and double click on it after it appears in my search result, and open the 'Groups', I can't see the user group.
Are you logged in to the instance as an admin ? If not please use admin to do so and verify the behavior. If still not able to perform the task enable the browser debug tool as it may give you some hint on whats the issue. The classic UI is not very good to handle the errors returned from the server side so the process just dies without showing any error in UI.
I remember this sometimes happens with classic UI console. Could you please try the same with touch ui console. Plus with classic UI it wont show any proper error if and all any exception happened but the same is not case with touch ui.