Value selection from a list with regard to a GuiXT pop-up window:
I have a list of 36 items. What we require is a pushbutton that invokes a popup screen listing these 36 items. We then want the user to be able to select one of the items via a radio button; and the selected radio button value to populate a field. This can be done via HTML but we would prefer a popup instead, if you can please tell us how?

You may use the "/O" screen (Session overview) to create the popup:
Use a pushbutton that executes the function code "/O" and starts an InputScript.
In the InputScript, delete all elements of this screen

Screen rsm04000_alv.2000

  WindowSize (20,120)

  // delete all standard elements

  del (0,0) (20,120)

  del P[/0]

  del P[/5]

  del P[/12]

  del P[/14]

  Title "xxxx"

and generate the selection in form of radiobuttons or pushbuttons. Do not put an "Enter" here; the screen will then be shown as popup. After the user has made a choice you continue.

An alternative is to provide your own searchhelp for the InputField. In this case you write an ABAP program that displays the 36 values. See InputField documentation, ABAP exit program.