HBI supports order_by keyword for listing inventory groups, we have to support it too.
Following is some info from them. We can ignore the host_count 

groupOrderByParam: name: order_by in: query required: false schema: type: string enum: - name - host_count - updated description: Ordering field name hostOrderHowParam: in: query name: order_how required: false schema: $ref: '#/components/schemas/OrderHow' description: >- Direction of the ordering (case-insensitive); defaults to ASC for display_name, and to DESC for updated and operating_system

host_count is out of scope: https://redhat-internal.slack.com/archives/C08GTNAFZ55/p1760012336164719