We have a problem with two simple scripts. 
step 1 : You go in the first screen of VA01.
step 2 : Then you click on Back green icon (/3), so you are on the SAP Easy Access where there is no script. 
step 3 : And then you click on Log off yellow icon (/15). GuiXT returns to VA01.
How can we rectify this very strange phenomenon?

Please put the statement

Disable auto return

into the InputScript.

Explanation:
If the processing of an InputScript ends with a new transaction, GuiXT interprets the next F15 key as a return to the original transaction where the InputScript started. This makes sense in many cases, but not in a case where the InputScript itself already realizes a "backwards" functionality, as it is in your case. With "Disable auto return" the standard F15 remains valid after the InputScript processing.