Can the SQL Server optimizer use indexes to update auto created statistics?

Consider the scenario where auto created statistics are created on a column to satisfy a query.

Afterwards; a nonclustered index is created on the column, with or without other key columns.

Could the SQL Server optimizer leverage indexes to update these auto created statistics when running UPDATE STATISTICS statements?

Is there a difference between (default) sample rates & fullscan statistic updates when adding these indexes?

SELECT @@VERSION;Microsoft SQL Server 2017 (RTM-CU12) (KB4464082) - 14.0.3045.24 (X64)   Oct 18 2018 23:11:05   Copyright (C) 2017 Microsoft Corporation  Developer Edition (64-bit) on Windows Server 2016 Datacenter 10.0 <X64> (Build 14393: ) (Hypervisor)