Italy didn’t hold on to British Somalia for very long. Britain recaptured it in early 1941, and likewise liberated Ethiopia and conquered Italian Somaliland. Germany didn’t save Italy’s rear end in that part of Africa, nor did it even intervene as far as I know; when Germany sent the Afrika Korps to help the Italians in March 1941, it was to bolster their position in North Africa, not in the Horn of Africa (which the Italins had already lost by then).
Italy has been described (by Gordon Prange, as I recall) as “the tail of the Axis kite”, but Mussolini wasn’t a subcontractor whose job was to take orders from Hitler. I don’t think there was a lot of coordination of war aims or activities between Berlin and Rome; Hitler certainly did have to bail out Mussolini when the latter’s military adventures in Greece and North Africa went badly wrong, but part of the reasons those misadventures happened in the first place was that Mussolini was playing his own game rather than coordinating his actions as a member of an Axis team. He probably would not have appreciated being told by Berlin what countries he could or could not invade, and his reaction would probably have been as arrogantly contemptuous as the reaction of Benzino Napolini in the Chaplin comedy The Great Dictator when Adenoid Hynkel tells him to keep out of “Osterlich” (which Hynkel himself wants to invade).