Hi,
Responsibility for Operating systems is on the cloud provider only for Platform as a Service (PaaS) and Software as a Service (SaaS). For Infrastructure as a Service (IaaS) and On Premises it is customer's responsibility.
For example, if you create a VM in Azure the responsibility for the operating system is yours since that is IaaS. On the other hand, if you host a website using Azure App Services the responsibility for the operating system is on Microsoft since Azure App Services is PaaS.
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Thanks.
-TP