I'm pretty sure it was just Mussolini as I doubt most Italians even cared about Ethiopia prior to 1935. I'm not completely sure but Italy might have come after Ethiopia if the First World War hadn't happened or if they had remained neutral during World War 1 but I kinda doubt that as well.
Italian Eritrea was considered one of, if not the most developed European colony in Africa whose growing Italian community made up roughly 12-14% of the entire Eritrean population by the 30s-40s if Italy had been given enough time to develop. Italian settlement in Eritrea (and maybe Somaliland) is going to be popular whereas not many people are going to want to go to Ethiopia as the continued insurgency and lack of resources didn't make it appealing but you might see a significant Italian minority if the Italians somehow managed to hold onto it until the late 40s or 50s. I seriously doubt the Italians are going to be able to completely pacify Ethiopia but if it can curb the resistance to an acceptable level, the Italians might be able to bring in more settlers and develop Ethiopia's flourishing urban centers.
Yeah, the Allies are going to do anything to secure Italian neutrality during the Second World War which may include British or French Somaliland and French Tunisia being ceded to Italian control in return for no Axis Italy or a neutral Italy. The Ethiopian population was around 16-17 million and not around what, 780,000 Libyan Arabs - the Italians will not be able to push off the natives without some significant action that will inevitably lead to a boost in the numbers and popularity of the Ethiopian Patriots.
Then I tend to see Italy forgo Ethiopia and focus on the existing colonies. Mussolini had sense enough to oppose Germany in rebuilding its position and that puts him in the British camp unless they drop the ball. Ethiopia looks like a foolish Mussolini power play that can be butterflied away.
Now I am not certain the Italians push off the native Libyans without the impetus of Fascist callousness but in this era it is not overly worried about displacing natives so I leave Libya a bit vague. But I concede that as the "Fourth" shore it takes little to hold Italy's interest and slowly settle the coast, strategically it deserves more attention, so I will concede it is holding out until oil changes the math. Italy should be focused on Eritrea and next Somilialand, and so long as colonialism has legs Italy should hold them. And I would bet Italians swell to outnumber Libyans, at least 2 to 1, maybe 3 to 1, Libya is Algeria in reverse here, a part of Italy with a minority or the coast carved off and vassal to the South, either way Italy is in possession, maybe altering the way France reacts in North Africa.