Idea came from StackExchange:  What happens to feature class when importing to feature dataset?

From Esri documentation:  About importing feature classes.

Suppose you want to import a feature class that's in another coordinate system. For example, you may want to import a feature class that uses the North American Datum (NAD) 1927 coordinate system into a feature dataset that uses the NAD 1983 coordinate system. To import the feature class, use the Project tool to simultaneously import the data and convert it into the new coordinate system.

 

Learn more about the Project tool reference

 

If the feature class you're importing is already in the coordinate system you want it to be in for the geodatabase, use the Feature Class To Feature Class or Feature Class To Geodatabase tools to import the data. The feature class you create with these tools can standalone or import into an existing feature dataset.