How to have rounded (beveled?) edges in inset? As per the attachment, I'd like to have rounded (beveled) edges in the places marked, and then inwardly extrude the face in question (or can extrude first, then round the inset face). Many thanks in advance if someone can tell me how to do this. 

I tried using edge loops, scaling the loops out towards the edges of the inset face, and then scaling down the edges with proportional editing, but that didn't work.
You can effectively do it extruding first.


Inset I
Extrude inward E
Select the inner edges
Bevel CtrlB and tune the amount of bevel edges with the mouse wheel or with the numpad + and -


In 2.79 (as you have here)



Or in 2.8 you can also use the new tools (keyboard short cut are still available, though)