In my InputScript, I start a new mode with /O and would like to change the value of an InputField in the mode I started. But a "Set V[...]" command does not work. What  is wrong with this approach?

When you start a new SAP mode with /O, GuiXT uses a completely new set of variables for the new one. This means, V[enr1] in one mode is different from V[enr1] in another mode.

I suggest you either use the  "Returnvalues" command (please see the keyword documentation for details), or you use shared text variables as described in the "CopyText" documentation, options "toSharedText" and "fromSharedText".