The allies did not give Italian Somaliland back, per se. Italy lost her colonies as she signed the Paris Treaty in 1947.

Rather, the United Kingdom administered the former Italian Somaliland from 1942 until 1949, when the UN granted Italy a 10 year trusteeship of the territory, with provisions for Somali independence in 1960.

It is fair to say that the Trusteeship system, and the particular decisions of how to handle the former Italian colonies was a particular concern for British and American diplomats in the period from 1945 to 1949. There were calls within sections of the British government to try and retain the Italian Somaliland territory as a British territory, while Clement Atlee as Foreign Minister called it an "incubus" and predicted it would be an economic drain on the empire.

From the American perspective, it was important to protect the Wilsonian ideal of self-determination of nations. The American post-war position was to champion a gradual move towards independence for these Italian colonies, and to prevent them from simply becoming British or French colonies. Furthermore, after a Soviet proposal for a British/US/Soviet joint trusteeship of Somalia, American diplomatic efforts began to see the merit of Italian trusteeship as a way to avoid Soviet influence spreading to the Horn of Africa.

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Source

1- Reviglio della Venera, M. 2014. The United Nations, Italy and Somalia: A 'Sui Generis' Relation (master's thesis). Retrieved from Utrecht University Repository pp 24-31.