The question is in the title.
After Italy took over Ethiopia before the war did they do anything to fight in the war on the Axis side? Did they send troops to fight for Italy? Did they support the Italian war machine in any way?
Four words: Regio Corpo Truppe Coloniali
Of the 256,000 Askaris in the Italian Royal Army, 182,000 were from Italian East Africa (I don’t know how many of them were Ethiopians though; Somalis were also recruited).
As of supporting the Italian war effort, among the general Ethiopian population, most of them were against Italy (for obvious reasons). The Ethiopian resistance (Arbegnoch) was supportive of the British war effort against Italy.